What is the current measured (i.e. model independent) density of hydrogen $H$ in the universe? The question came up in this thread: What is the current value for the temperature at which Recombination took place? I was under the impression that we had an independent measurement of Hydrogen in the current universe in order to test out the predictions of BBN.  One of the respondents seems to be suggesting that there is no such measurement and that we need to have a model of Cosmology before we can estimate the average density of H and He.
So is there a current measurement of the $H$ (neutral and ionized) density that doesn't rely on the assumptions of $\Lambda $CDM?  If not, how do we know that the prediction of BBN of ~75% is correct?
 A: There are only lower limits to the measured baryon density in the local universe.
The baryon density should be about 4.6% of the critical density of the universe (about 6 H atoms per cubic metre). This value arises from modelling of the cosmic microwave background, but before that it had already been estimated at around that level from determinations of the primordial helium and deuterium abundances.
The amount of luminous baryonic matter in the form of stars is much less than 1% of the critical density; for a long time there has been a second "dark matter problem" - that most of the baryonic matter was undetected.
There has however been much progress in finding hot gas in clusters of galaxies and warm gas in halos around galaxies and in filamentary structures between galaxies. This warm-hot intergalactic medium may account for 40% of the required baryons and it has been claimed recently that it is sufficient to account for all the "missing baryons" (Nicastro et al. 2018).
An excellent review of the topic is provided by Shull et al. (2012) who conclude that the current census (see picture below) gets to within $29\pm 13$% of the value suggested by the CMB, but I think it is fair to say that others would regard an uncertainty of this size as highly optimistic.
Pie chart showing the likely contributions to a whole pie that represents the 4.6% of the critical density suggested for baryons by CMB measurements (Shull et al. 2012).

Note that most of these measurements depend linearly (or in some cases, to the power 1.5) on the current Hubble parameter, since this is used to estimate (large) distances and volumes.
As a final comment, I would note that the "model dependence" of the 4.6% number boils down to the factor by which we think the universe has expanded since (a) the epoch of primordial nucleosynthesis and (b) the epoch of recombination, since our measurements of those phenomena yield estimates of the baryon densities at those (very different) epochs, and baryon number is conserved. 
