Changing a tyre would not require any work near the cambelt, so regardless of the tyre shop's competence, they didn't cause the timing issue.
A proper cambelt job must include tensioner. These tensioners are a bit fiddly to set up, so I'm guessing they got that wrong. There's not much space, and has to be done with a mirror. Holding a mirror, and a tensioning tool, whilst with your third hand, tightening the tensioner, requires care and dexterity. Standard stuff for most mechanics. The more difficult it is, the more care you take.
They cocked it up, now hopefully they've fixed it, and learnt.
Whilst re-threading a damaged hole can be effective for many things, I would be wary for a wheel bolt. An alternative would be to fit a thread insert (helicoil), but again, not for me with wheels.
I agree, most unlikely anything done on a tyre change would cause the engine problems.
Cambelt drive trains include the belt itself, the tensioner, maybe some idlers, maybe a water pump. The most common failure is the water pump bearings due to seals breaking down and allowing coolant to wash out the lubricant which causes the bearings to fail. Next most common is idler and tensioner bearings. As a result of these components failing the belt can easily be damaged and, although the water pump or idler/tensioner bearings may well be at the root of it it's the poor old belt which gets the bad name. So refitting a used idler/tensioner is a really silly thing to do unless it's known to have been renewed very recently for, perhaps, some other reason (unlikely). Belts do fail, but far less frequently than their reputation would have you believe.
So, yes, I too think they "cocked it up" Hopefully now the whole thing has been put back together more carefully and all should be well.
Tyre shop may have damaged it. Or a previous tyre shop. Can't point fingers there, unless the whole history is known.
Tyre stores are notorious for using air (or electric) impact wrenches to remove and tighten wheel nuts. I'm well known in my area as a "man who knows, or at least fiddles, with cars" and I get maybe 2 or 3 neighbours every year knocking on my door because they can't undo their wheel nuts. It used to be worse but has improved of late because some workshops are now using torque wrenches for final tightening, but many still just whack the gun on and let it rattle away at the nuts. This practice stretches the threads in the hub and, after a number of times of overtightening, you end up with the problem you have. Of course the other bolts will have been tightened in the same manner so may also be stretched. A new hub flange is the best solution but an engineered thread insert like a "time sert" would probably work quite well and be cheaper to do. I wouldn't feel so happy with an insert of the "wire" type - as mentioned above - because they are not so good for applications where frequent dismantling and reassembly are expected (like a wheel retainer). By the way, replace the retaining studs/bolts/nuts too as their stretched threads will be likely to weaken the new hub threads.
Tyre stores are blamed for a lot of things amongst which are the above but also poor use of jacks causing damage to sill seams and rustproof undercoatings. Of course, as with many things in this life, there are good ones and bad ones. I don't let my vehicles get anywhere near them. Until early this year I've been well known, and laughed at, for removing family vehicle wheels on my driveway, sticking them in my wheel barrow and wheeling them the mile up the road to the small tyre specialist whose garage is there. That way I get to do the jacking up and retightening of the wheel nuts myself so avoiding body damage and ensuring correct tightening of the wheel nuts. I've now circumvented the transportation problem by using a mobile tyre fitter who comes to my house, i jack the car up and remove the wheels, he does the tyres and wheel balancing and I then refit them - I can thoroughly recommend this to anyone.
If you're going to a tyre place one wee hint is to watch how they remove the wheel nuts/studs. Do they use a power wrench on the security key for the security wheel nut/stud? This is extremely bad practice as it will ruin the security scroll after a few times of doing this. If they use a manual tool - power bar or similar- you just may have someone who knows what they are doing! By the way, it's perfectly OK to remove the other hex headed bolts/nuts with a power gun and OK to "snug down" the nuts/studs with a power gun when reassembling but the gun should be on low power - which they never are - and final tightening should be done by torque wrench.
