At the risk of self-plagiarism: (https://physics.stackexchange.com/a/532568/43351 with a bit added).
Molecular chemistry in the early universe requires species with bound electrons. Helium hydride is the first molecule to form because neutral helium atoms, formed about 120,000 years after the big bang, could combine with plentiful protons; but it was another 260,000 years until significant numbers of neutral hydrogen atoms formed, and it is only once these are present that there is a route to forming H$_2$.
Details:
Helium has a much higher ionisation energy than hydrogen and therefore starts to recombine at higher temperatures (about 7000 K at redshifts of $\sim 2500$, compared with 3000 K and a redshift of 1100 for hydrogen). Thus in the primordial gas of hydrogen and helium, it is the helium that recombines first. There is therefore a period of time in the early universe, $120,000 < \tau < 380,000$ years, in which almost all the hydrogen is ionised, but most of the helium is in the form of atoms.
The two react to form helium hydride
$${\rm He} + {\rm H}^{+} \rightarrow {\rm He H^+}$$
As you might expect, the concentration of this molecule is low, because the temperatures were still high enough to easily radiatively disassociate it - about 1 part in $10^{21}$ at $z \sim 2000$ (Stancil et al. 1998; Galli & Palla 2013).
This is the first molecule to be produced with any important level of abundance. Note, this appears to be a somewhat arbitrary definition, since it is also claimed (e.g. Lepp et al. 2002), that He$_{2}^{+}$ formed the first molecular bond, via He$^+ +$ He, but was too weakly bound to survive in any concentration (the concentration peaks about 100 times lower than HeH$^+$ according to Galli & Palla 2013).
It is also possible to form small quantities of $H_2$ at $z> 2000$ via the reaction of a hydrogen atom with another in an excited state: H + H$^* \rightarrow$ H$_2$; but of course, although the hydrogen molecule is much more strongly bound than HeH$+$ (the dissociation energy of $H_2$ is 4.5 eV, versus about 1.8 eV for HeH$^+$), there is very little atomic hydrogen present and this reaction requires not one, but two hydrogen atoms to get together. It is not until H atoms recombine in quantity some 260,000 years later that hydrogen molecules are formed in various gas phase processes and H$_2$ becomes the dominant molecular species.
The dihydrogen cation, H$_2^{+}$ cannot form directly before either helium hydride or atomic hydrogen are formed as precursors. e.g.
$$ {\rm HeH}^{+} + {\rm H} \rightarrow {\rm H}_2^{+} + {\rm He}$$
$$ {\rm H} + {\rm H}^+ \rightarrow {\rm H}_2^{+}$$
and thus H$_2^{+}$ forms after HeH$^+$.
