Hypothetical tree damage

[FlamedToACrisp]

I had my tree trimmed down about 1.5 years ago, but it's grown again and is pretty big.

It's on my boundary and is overhanging the neighbour's parked car. Hypothetically, if a branch broke off in a gale and landed on his car, who would pay - my neighbour's car insurance or my house insurance?

We had a huge tree in our backgarden, so we did talk with insurers.

If our tree had fallen and landed on their house, their insurers would have to pay.
We would not have been liable unless we hadn't kept the tree maintained.

However, there's two things here from your question. Firstly, I assume it's overhanging the neighbour's property. Secondly are you thinking a specific branch might, or just a general question?
It might well be different if your tree is overhanging another property and that comes down. Secondly, if the neighbour has said to you that they think it might come down, (especially in writing so they have proof) or f there is a branch that looks like it might come down, then you may find yourself liable as you should have kept that maintained.

It is overhanging, but I can't just have one side trimmed, it would unbalance the tree. No specific dodgy branch, but last time we had strong winds, a branch blew off and hit my car. Neighbour hasn't commented on the tree.

It shouldn’t be overhanging at all. Is his car parked on his property or on the public road?

Other than chopping off the whole of the leafy part of the tree and leaving nothing but a stump, there is no way of it NOT overhanging - like I said, it is literally on the boundary.

His car is on a parking area shared with other neighbours, but not with me. I have my own parking space, under the other side of the tree (I've moved my car onto the lawn tonight).

This is why you don’t plant large trees on the boundary, someone else gets half... I appreciate that it’s fully grown by now, but you have no actual entitlement to grow stuff over someone else’s property.
You will be liable for any damage caused.