Speed of birth of a star A star forms when enough material gets close enough. But how fast does the star start to shine?
Is it gradually heating up (similar to an oven) slowly shining more and more?
Or is it more like an explosion (similar to an atom bomb) that happens the second the critical mass is reached?
 A: Assuming that by "shining" you simply mean "emitting light", the answer is that it starts slowly and gradually.
Normally, when we think of a star shining, it is a hydrogen-burning star on the Main Sequence, where any star will spend the majority of its life time. But stars have significant light emission well before hydrogen fusion sets in and they settle on the main sequence. In fact, most lower-mass stars are more luminous before they settle on the Main Sequence. 
When the matter of the stars contract, the gravitational potential energy released by the contraction becomes thermal energy, heating up the protostellar matter which starts shining with a blackbody spectrum corresponding to its temperature. This will initially have its peak intensity in the infrared, but as the star heats up, the peak intensity (defining the apparent color of the star) will move towards the optical.
However, not all the matter in the protostellar cloud actually collapses into the star. A significant fraction stays behind as a thick, dense and opaque envelope surrounding the star and blocking its radiation (except in the mid-far infrared), until the temperature of the protostar rises top a point where its energy output will disperse the enveloping cloud, and the stars becomes visible as what is known as a T Tauri Star. These stars are very luminous, but do not have any internal fusion processes (except possibly, Lithium fusion at the very latest stages). 
To see how stars typically develop before hydrogen fusion sets in, here is a link to Wikipedias rendition of the so-called Hayashi- and Heyney-tracks which show the development in luminosity and temperature of Young Stellar Objects (YSOs):

The blue tracks describe the evolution of stars of different mass from they shed their envelopes (the so called Birth Line, upper black line) until they settle on the (Zero Age) Main Sequence (lower black line). The numbers in blue show the mass of the star tracked in units of the Sun's mass. The red lines are isochrones, showing at which times the different tracks have reached a certain point of development. These can help us see that more massive stars, e.g. a 6 $M_{\odot}$ star, is almost through its YSO evolution at a time of $10^5$ years, where lower mass stars have hardly shaken off their surrounding envelope yet.
It is interesting to see that all stars up to around $1.5 M_{\odot}$ are less luminous when they have started "shining" (i.e. fusing Hydrogen) than when they left the Birth Line. For the lowest mass stars, the luminosity drops by several orders of magnitude along the vertical Hayashi tracks. They have essentially the same surface temperature and hence surface brightness (brightness per surface area), but due to contraction have a significantly decreasing surface area. 
Higher mass stars act the opposite way: they keep roughly the same luminosity, but as they contract, this will have to escape through a smaller surface, which gives a significantly higher surface temperature, showing as a roughly horizontal evolutionary track on the HR diagram. But even these lose a bit of luminosity directly before the onset of Hydrogen burning.
Intermediate stars follow first a vertical Hayashi-, then a horizontal Heyney track, the onset of the latter depending on the mass of the star.
At the Zero Age Main Sequence, the stars start fusing Hydrogen, which stops the contraction of the star and stabilizes the balance between gravitational infall and radiation pressure from within and lets the star settle into a steady state at the Main Sequence. But as can be seen, the main sequence does not mean that the shining starts; rather, Hydrogen fusion stabilizes the star and provides fuel for it to stay shining for billions of years (for the lower mass ones), albeit at lower luminosity than the initial heat due to gravitational collapse could provide.
A: Summary: There is no explosion at the birth of a star. It is a
gradual process. The star first heats from the potential energy used
by matter collapse, and starts radiating like any hot object. When the
core temperature reaches some 10 millions debrees K, the fusion
reaction starts. The energy liberated stops the collapse, and takes
over the heating of the star at a core temperature which is on the
same order as that finally produced by the collapse. There is no
explosion as the process is contained by the pressure of surrounding
matter. There is no sudden increase of temperature either at the
surface as it is dampened by the matter surrounding the core that is
not participating in the fusion except for providing pressure. The
increase of temperature is also a continuing process during star life
as heavier elements produced by fusion increase density and pressure
in the core.
.
The stars form from the collapse of a large cloud of dust and gas. The
potential energy from the collapse heats the collapsing matter. At a
sufficient level of pressure (as density increases) and temperature,
the star starts producing energy by nuclear fusion of
hydrogen. Logically it should start at the center of the star where
temperature and pressure are maximum. Indeed, stellar evolution is by
fusing matter from the core out, so that near the end of its life the
element composing the star are in layers from the lighter in the outer
layer to the heavier at the core.
Since the star heats up before "igniting", I guess it is radiating
more and more. I do not know whether that counts as shining, but
it is visible. The energy radiated is proportional to the 4th power of
the temperature (Stefan-Boltzmann equation).
The fusion of hydrogen to deuterium then helium starts when the
temperature in the core reaches 10 millions degrees. But the the
temperature of the outer layer is much less.
It is currently around 6000K
for the sun photosphere for a core temperature of 15 millions
degrees K, with the envelope (60% of the mass) in between cooling
progressively towards the surface.
The fusion reaction takes place in the hot core of the star, then the
energy must travel through the rest of the star matter before shining
outside. In the sun, and I guess in young stars, the journey to the
surface is first radiative, then convective. So it must take some
time. I would also think that the matter in the envelope has to be
heated further before tranmitting the energy and radiating (though the
sun was already quite hot from collapse before fusion started). So
that must be progressive, but I do not know the figures. It
most likely depends on the mass of the star. This heating of the
envelope may be seen as taking the relay from the collapse heating,
though it is probably faster. Since the core was much hotter than the
surface even before fusion started, it is likely that the convection
cells are already formed to carry the heat when fusion started.
But I do not think this should be seen as an explosion. It is
contained by the pressure, which it actually fights as long as the
star lives. So the fusion reaction stops the collapse and take over
for heating the star.
After that, the new star is shining, but its shining increases with
time. The sun's shining increased by 30% since its birth 4.5 billion
years ago. This is due to a rising core temperature that could be caused by
a rising core density and pressure as the proportion of heavier helium
nuclei increases.
To conclude I would think that given the mechanisms invloved in the
lighting and evolution of a star, it must heat gradually, though there
may be some variations.  However I took most of my information from
the evolution of the sun, and I have no idea whether the mode of
lighting is the same for very different masses.
Note: this is a personnal conclusion from the information I knew or found on the web.  I did not find the direct answer to the question in an existing document.
