About what 'time' in the Universe's history did the r-process and s-process begin respectively? I was reading about this but there is something for which I haven't found a reliable source yet. 
When did each process begin and is there any estimation of the abundances of the elements throughout the history of the Universe?
Thanks in advance.
 A: 
Q: "About what 'time' in the Universe's history did the r-process and s-process begin respectively?"

The big bang occurred 13.799 ± 0.021 Gya. Approximately 9.8 Gya the s-process started, during the formation of the first population stars. The r-process was approximately 14 billion years ago.

Q: "... estimation of the abundances of the elements throughout the history of the Universe?"

An image included at the end of this answer shows how the stars are divided into "populations" and the elements in each category, with Pop III (the oldest) containing the lightest elements and Pop I the heavier metals.
The birth and death of the earliest stars lead to the next generation and after the second population the r and s processes started.
The paper "Nucleosynthesis of Heavy Elements by Neutron Capture" linked below lists "estimation of the abundances of the elements" but obviously does not show which star exploded when, nor how much of each element was released at a particular moment in history.
Once the stars started to form so did some of the heavier elements, it's an ongoing process of creation and destruction.

Supporting information and links for the answer:

Three Four basic processes can be identified by which heavy nuclei can be built by the continuous addition of protons or neutrons [mine: to seed isotopes]:
• p-process (proton)
• rp-process (rapid proton capture process)
• s-process (slow neutron)
• r-process (rapid neutron) 
Capture of protons on light nuclei tend to produce only proton-rich nuclei. Capture of neutrons on light nuclei produce neutron-rich nuclei, but which nuclei are produced depends upon the rate at which neutrons are added. Slow capture produces nuclei near the valley of beta stability, while rapid capture (i.e., rapid compared to typical beta-decay timescales) initially produces very neutron-rich radioactive nuclei that eventually betadecay towards the valley of beta stability. Some nuclei can be built by more than one process.
These are the only two ways nature can assemble heavy elements. We should not be surprised then that there are two major distributions of heavy nuclei: the r-process and the s-process distributions. We shall see that the r-nuclei in our Solar System likely formed in an environment that experienced a freezeout from equilibrium while the s-nuclei must have formed in an environment that was striving for, but never reached, NSE. The differing character of these scenarios results in the different character of the r- and s-process abundance distributions.
Once an abundance of heavy elements is available, nature may make modifications to it by exposing it to a flux of photons, neutrinos, or nucleons. Such events are probably responsible for the production of the majority of p-nuclei. 

[ The above introduction was sourced from "Nucleosynthesis: the s-, r- and p- processes" and the following paper. ]
There are various papers upon the subject, the difficulty of this subject is well explained in the paper "The r-, s-, and p-processes in Nucleosynthesis" by Bradley S. Meyer where he writes: "It is impossible for a single paper to cover all relevant aspects of the r-, s-, and p-processes ...".
The paper "Nucleosynthesis of Heavy Elements by Neutron Capture" attempts to analyse some of the previous work and determine when the r and s processes began, due to the variety of means to initate the creation of these processes it's difficult to nail down an $\underline {exact}$ time.
On page 3 they write:

"... The remainder of the paper will be concerned with the implications of these abundances for ideas concerning nucleosynthesis. The elements iron, cobalt, and nickel, excluded from Table 1, provide a different class of abundance problem, both experimentally and theoretically. Although they are syn- thesized primarily by a different process (e-process) than the heavy elements whose abundances constitute the main burden of this paper, they have significance as seed nuclei for the s- and r-processes. The ^-process certainly has built upon seed-iron nuclei, and the r-process may have done so as well. The fraction of iron-group nuclei exposed to neutron fluxes of various intensities has important implications for nucleosynthesis and galactic history."

Thus the paper concludes on page 45:

"This result is in substantial agreement with the conclusions of Hoyle and Fowler (1963), to which the reader is referred for the detailed consequences of assigning the r-process to massive stars which evolve rapidly and are thus capable of producing r- process elements early in the history of the Galaxy as well as subsequently. 
The uncertainty indicated in equation (46) is an estimate based on the range of conditions which were found in Section Va to be suitable for production of r-process material in good agreement with the observed abundances of the r-nuclei. It is of interest to note that stars with masses in the range given by equation (46) are just those which Iben (1963) and Fowler (1964) suggest may well be disintegrated by explosive nuclear burning after the onset of general relativistic instability. If the third r-process peak was synthesized in the same object, the relation p ~ $T_{9^3}$ from equation (44) can be compared with Figure 14 to find the conditions corresponding to a solution with cycle time 3 sec; the results are $T_9$ = 1.0, log n$_n$ = 25.5 as shown in Figure 18. 
It becomes increasingly important to have some independent evidence for locating the r-process site, whether massive objects, conventional supernovae, or both.".

The theory of Big Bang Nucleosynthesis states:

"It is now known that the elements observed in the Universe were created in either of two ways. Light elements (namely deuterium, helium, and lithium) were produced in the first few minutes of the Big Bang, while elements heavier than helium are thought to have their origins in the interiors of stars which formed much later in the history of the Universe. Both theory and observation lead astronomers to believe this to be the case. 



*

*Recent Work:
In the paper "Origin of the heavy elements in binary neutron-star mergers from a gravitational wave event" by Daniel Kasen, Brian Metzger, Jennifer Barnes, Eliot Quataert, Enrico Ramirez-Ruiz (Submitted on 16 Oct 2017) they write:

"Here we report models that predict the detailed electromagnetic emission of kilonovae and enable the mass, velocity and composition of ejecta to be derived from the observations. We compare the models to the optical and infrared radiation associated with GW170817 event to argue that the observed source is a kilonova. We infer the presence of two distinct components of ejecta, one composed primarily of light (atomic mass number less than 140) and one of heavy (atomic mass number greater than 140) r-process elements. Inferring the ejected mass and a merger rate from GW170817 implies that such mergers are a dominant mode of r-process production in the Universe.".

Elements heavier than Iron ($F_e$) and where they are thought to have been formed is shown in this chart from the Wikipedia Stellar Nucleosynthesis webpage:


"Big Bang nucleosynthesis produced no elements heavier than lithium, due to a bottleneck: the absence of a stable nucleus with 8 or 5 nucleons. This deficit of larger atoms also limited the amounts of lithium-7 produced during BBN. In stars, the bottleneck is passed by triple collisions of helium-4 nuclei, producing carbon (the triple-alpha process). However, this process is very slow and requires much higher densities, taking tens of thousands of years to convert a significant amount of helium to carbon in stars, and therefore it made a negligible contribution in the minutes following the Big Bang.

Thus, the reasoning for my answer, having established where heavy elements came from (first, excluding man-made) your question is when: The answer is: The r-process and s-processes first started during the formation of the first stars. 
References: (Thanks @Thomas)
"Observing the r-Process Signature in the Oldest Stars" by Frebel, Anna.
"R-process enrichment from a single event in an ancient dwarf galaxy" by  Ji, Alexander P., Frebel, Anna, Chiti, Anirudh, Simon, Joshua D.
"ACS Imaging of the Ultra-Faint Dwarf Galaxy Reticulum II: Age-Dating a Unique Nucleosynthetic Event" by Simon, Josh.

From Wikipedia: "The s-process in stars"

"The s-process is believed to occur mostly in asymptotic giant branch stars, seeded by iron nuclei left by a supernova during a previous generation of stars. In contrast to the r-process which is believed to occur over time scales of seconds in explosive environments, the s-process is believed to occur over time scales of thousands of years, passing decades between neutron captures. The extent to which the s-process moves up the elements in the chart of isotopes to higher mass numbers is essentially determined by the degree to which the star in question is able to produce neutrons. The quantitative yield is also proportional to the amount of iron in the star's initial abundance distribution".



"Stars observed in galaxies were originally divided into two populations by Walter Baade in the 1940s. Although a more refined means of classifying stellar populations has since been established (according to whether they are found in the thin disk, thick disk, halo or bulge of the galaxy), astronomers have continued to coarsely classify stars as either Population I (Pop I) or Population II (Pop II). They have even postulated a third population (Population III; Pop III), though stars of this type have yet to be observed.
The classification system is based on the metal content of the stars (their metallicity, usually given the symbol [Z/H]). Pop I stars are the most metal-rich, with metallicities ranging from approximately 1/10th to three times that of the Sun (i.e. from [Z/H]=-1.0 up to [Z/H]=+0.5). This means that the gas from which Pop I stars formed must have been recycled (incorporated into, and then expelled) from previous generations of stars a number of times, and that Pop I stars are relatively young compared to Pop II and Pop III stars.
The Sun ([Z/H]~1.6) is a fairly typical Pop I star, as are most of the stars in the immediate solar neighbourhood. In fact, the majority of the stars contained within the thin disks of galaxies are Pop I stars, but Pop I stars can also found in the bulge.". [Source: http://astronomy.swin.edu.au/cosmos/P/Population+I]



"Figure 1. Simple illustration of chemical enrichment of the universe: massive Population III stars form out of primordial gas, explode as supernovae, and enrich the interstellar medium with products of stellar nucleosynthesis. Subsequent cycles of star formation and death (Population II) steadily enrich the universe with metals over time. The rst low-mass stars to form in the universe are still observable today. Two main contributors to chemical enrichment after the rst stars are 8 − 10M⊙ stars that explode as core-collapse supernovae, and less massive stars that enrich the interstellar medium via strong mass loss and stellar winds (AGB stars). Their nucleosynthetic products, the r-process and s-process elements, are the subject of this review". [Source: "Observational nuclear astrophysics: neutron-capture element abundances in old, metal-poor stars"]

