Update
The car is fixed and we're flying out on Monday to drive it home. We didn't manage to get a loaner in the UK, so it's been problematic only having one car. Tesla emailed yesterday asking me to contact the Reading service centre to see if I could get a loaner, but it's a bit late now. (Isn't it also more or less impossible to actually talk to a service centre directly these days unless your car's in for a service?)
Tesla have agreed to go above what their terms state with regards to covering costs, although the exact amount isn't clear to me yet.
Total extra costs to me so far, before Tesla have covered anything are £2,115.18
Tesla has agreed to cover at least ~£700 of that, possibly more. (£235 of that is the recovery cost, that they always said they'd cover)