I've just checked the Chiltern Railway website (National Rail Enquiries don't show it) and there is a 21:53 train from Bordesley to Moor Street (2 minute journey apparently) which gives you 15 minutes to jog down to New Street to catch the 22:10. I'll be on that train hopefully but staying over...
I'm staying in the Wetherspoons hotel on Bennetts Hill outside New Street. Very handy for The Wellington and also to finish the night's drinking once the Wellington closes at midnight.
Another reason a Saturday fixture would have been preferable at St Andrews: Bordesley station is a 'Parliamentary' railway station in that it only has a very, very limited service (maybe one or two a day - enough to keep it functioning) but on matchdays they run trains to & from Moor Street...
Bus No 60 from opposite the away turnstiles (adj Arthur Street) leaves at 9:42, ride 18 mins to Pavilions then a 6 minute walk to BNS station. Tight but do-able
Edit: or pre-book a taxi to pick you up from outside the Ibis hotel on the roundabout