The logistics and economics of having more than one recovery vehicle depends on the frequency (and location) of breakdowns.

Most recovery jobs are not urgent (a replacement vehicle will have been sent in most cases).
Only when the casualty is in a dangerous position would the police insist on rapid recovery.

A well-run depot will not schedule breakdowns (though they will, of course happen), so keeping a recovery vehicle on standby will not be top of the list - more likely crews of fitters in vans with spares ready to rush out and rectify faults.

Some time ago I saw 1001 in Corbridge late at night trying to fix a Citaro Ten that had been parked at the bus stop for several hours with a major air leak (the compressor supply pipe had pulled through the olive) using very few tools and no spares rather than have to remove the drive shafts and wind-off the spring-brakes and suspend tow. The recovery driver said that he wasn't a fitter as such - just what he had picked up over the years. He struggled but he managed to fix the olive back onto the pipe so that the pipe didn't leak or blow off again and got someone out to drive it back.
A job well done!

I have seen Alpha recovering a (Classic?) AD122 from Hexham Wentworth (not this year obviously) doing a rear-end suspended tow - there were two 'fitters' in attendance on that occasion and they didn't hang about.