I've a friend who is a Registered Bat Roost Visitor, who quite frequently "fosters" bats so we get them visiting quite a lot. Funny little critters, there is something fundamentally scary about them, IMO. They just look, well, they look wrong. Generally friendly, though, they are unlikely to bite but they may do so if they are being handled. If he was bitten, then, he shouldn't have been playing with one as they are protected.
I don't think that the traditional "rabies" (the sort that dogs get) has ever been found in bats in the UK, and as it wasn't mentioned in the OP, I think that's an irrelevant distraction.
EBLV, as Arch posted, is pretty similar, though still very rare in the UK. I think only something like 9 cases have ever been found in the UK bat population and there have only ever been 4 cases of EBLV in humans across Europe. As every one of them has been fatal, you can probably discount that one from your uncle's neighbour's list as well.
There's plenty of diseases which can screw up the circulation, it's possible that he picked up one of them from a bat bite but the idea has to be not to go anywhere near a bat if you find one.
As an aside, my Batty mate has all sorts of detectors which can turn the noise which bats make into sounds audible by humans. It is fascinating listening to them zero in on their next snack.