Pay up mate and forget about it, ain't worth claiming after you get it insured.
Garage will not pay sod all - do they sell tyres by the way? Did you have any issues with them? would they have had reason to give you some stick, knowing you dropped it off and had gone and that they could say "well we tunred up at 8 and it was like that"
Just seems an odd time for a tyre slasher to be out and about, usually happens in the night, and all four done? any other cars about, did they get done.
Wont do any good, as no proof, but doubt i would give the garage the money for new tyres, would rather get a spare set of wheels on it and take it away after they have mot'd it