When did the first annular eclipse happen?
A quick and dirty estimate is as follows.

An angular eclipse occurs first if the Moon is at apogee and Earth at perihelion. Assuming that these two events coincide (which happens every ~8.85 years owing to lunar apsidal precession), an angular eclipse requires that ($a$ denoting orbital semi-major axis and $R$ radius)

$$
      \frac{R_\supset}{(1+e_\supset)a_\supset} &lt; \frac{R_\odot}{(1-e_\oplus)a_\oplus}
$$

or

$$
      a_\supset &gt; \frac{R_\supset}{R_\odot} \frac{1-e_\oplus}{1+e_\supset} a_\oplus\approx 348300\,\mathsf{km}
$$

where I assumed that the orbital eccentricities, Lunar and Solar radii, and Earth's semi-major axis didn't change (over the period considered here) and are given by
$e_\supset=0.0549$, $e_\oplus=0.0167$, $R_\supset=1737.1\,$km, $R_\odot=695510\,$km, and $a_\oplus=149.6\times10^6\,$km.

At the present we have $a_\supset=384400\,$km. If we assume an average annual change by 22 mm, we find a time of 

$$1.6\times10^9\,\mathsf{years},$$

though it could have been more recent if the annual change of $a_\supset$ was larger some hundred million years ago (which may well have been the case).

In a similar way, one may also estimate when the last full Solar eclipse will occur. This time, the Moon is at perigee and Earth at apohelion, when

$$
      a_\supset &lt; \frac{R_\supset}{R_\odot} \frac{1+e_\supset}{1-e_\oplus} a_\oplus\approx 400800\,\mathsf{km}
$$

which last occurs in $430\times10^6$ or $745\times10^6$ years, depending on whether the current lunar recession rate of 38mm/yr or the long-term average of 22mm/yr is used.