Interesting question. Technically the garage is 100% responsible and will need to claim via their liability insurance, however I suspect that’s void as they did not have it inside their compound which I bet is a term of their insurance, so you’d be stuck in a fight between the insurers and the garage. Their insurers have no liability to you, so if they refuse a payout then you have to chase the garage via court with all the risks that takes.
Your own insurer would be well within their rights to play hardball, but it’s complicated by the fact they chose where to send the car so have a duty to you. It’s a shame really, as that would be a nice way to avoid a claim and your excess, but hey ho.