Admiral will want you to use their "approved" repairer, as they did with me and my brand new Lexus when I had a bump. The approved repairer would never have worked on one (brand new model to the UK) and I flatly refused for anyone other than my Lexus dealer to do the work. I eventually won but you will have to roll your sleeves up. This is contrast to the new Tesla link with Direct Line, where the "approved repairer" for Tesla cars is Tesla, so no arguments or battles about extortionate parts/labour costs.
As far as the battery is concerned, that's not a worry, it is fully covered just like any other part. The concerns aired about battery coverage comes from, I believe, EV's where the battery is under a separate lease or rented from the manufacturer. Obviously not an issue on a Tesla.
One final point to bear in mind re Admiral (and most others) is that the policy is a "market value" policy. If you have a major prang and the car is a total loss you will get the market value, not the replacement value. The car loses X% the minute you take delivery so beware of the effective loss of some/all your deposit, and potentially more outstanding on finance than the car is worth. Some Insurers offer a "new for old" replacement in the first 12 months, (e.g. Aviva) but only if the new car of the same spec is available in the UK within 21 days. That doesn't work well with a Tesla as we all know!
Direct Line understand this and will offer the cash equivalent of a new same spec car, and you go and do what you like with it. Given the price rises we will all have seen from order date to delivery (not to mention the Jan rise on its' way), this was important to me. You can of course also buy gap insurance which I am probably going to do to cover me from month 13, which reduces the cost of that cover.
For full disclosure purposes I have no connection or affiliation of any kind with Direct Line.
I would ask for a copy of the policy terms and conditions, before purchase, and read them very carefully with a firm understanding of how Tesla cars don't fit into the mainstream UK insurance market.