I picked up a Bantam in a charity shop of all places, many years ago while on holiday in the Lakes.
It refused to start until I had the very worn small end bush replaced by a local engineer, after that it was a very reliable bike for pottering around on.
No surprise that a worn small end would cause problems but I didn't expect it to stop the bike from starting at all, I'd expect a worn big end would be even more of a problem that should be fixed as a priority.
New tyres are sensible if it's been standing a while. Good luck with it, have fun