You leave yourself at a disadvantage when your mechanic has vehicle, and money, but you have no written contract specifying completion date and important phrases like "time is of the essence". I know it's easy to take a relaxed attitude about this kind of thing when looking for independent mechanic work on the cheap, but years ago I also had a vehicle stuck in limbo for a couple months.
Was there a starter problem when you took it to him? Seems odd to have this develop with a vehicle just sitting there, unless you bought it non-running, in an unknown state, or if the mechanic is just an idiot and the only problem is it sat for a month and the battery is now too low to crank it.
As far as delays for the starter, I don't buy that as more than a few days delay (is this weekend work for him or his main source of income, at a repair shop?) since there are probably a dozen places within driving distance (or Amazon, Rock Auto, etc) where a starter could be sourced, and it's not enough money to (necessarily) need a deposit for, starters are about $60 and solenoid $20 (or less online) and it only takes between 15 min and a few hours at most to install.
I'd get into a conversation about the starter, find out what is going on there in case the mechanic thinks this is a blank check scenario where he's just going to hold onto the vehicle until you pay him whatever he demands... but I don't know the guy, he could be quite reasonable.
So far as liability, this gets into a gray area. Technically you don't owe him anything if he didn't get the work done to a usable state, but if he can demonstrate in court that the amount of work he did do, reduces your expenses to get the job finished then you own him that much, possibly minus the tow charge you wouldn't have incurred to move it a 2nd time to a different mechanic. If he does give you a refund on your deposit that might not be enough money to bother pursuing by itself unless you want to teach the guy a lesson. Again I don't know the guy, but it seems like you're being brushed off for him not to keep you updated on what is taking so long.
You are certainly allowed to show up and take your vehicle back, with title in hand and a sheriff or marshall accompanying you if he won't release it otherwise but then he could put a mechanic's lien against the title and sue for some amount, but of course he's hard pressed to win a suit if the work isn't completed.
You really should have gotten all this in writing, particularly with the 1 year warranty as that won't hold up in court at all from a verbal agreement if he just denies that, unless this is a repair business and you have written statements, photo of sign, brochure or something, that this kind of work carries such a warranty.
Anyway if this is his main work, source of income then I agree that the end of the week is a reasonable limit, but if he is doing it in his spare time and just needs a kick in the pants to actually start, two weeks might be my limit before showing up with a flatbed to take it as that allows for shipping time if parts need ordered, OR if you think he's the type to hold a grudge, not telling him anything in case he might sabotage the vehicle, just showing up as a surprise to take it. Again, I don't know the guy or his circumstances, for all I know it could be 15F degrees there with 3 feet of snow.
