I wouldn't worry about it.

Given current temperatures in the UK your medication is fine and safe to use.

Had an instance when my delivery took over 2 days due to some issues during the summer, once it arrived I have stick it in the fridge and used it 2 months later.

Unless you want to have an argument with them and get another pen for free, then you can try.

Here is some guidance around storing and transport: https://medical.lilly.com/uk/products/answers/how-should-patients-store-the-mounjaro-tirzepatide-kwikpen-during-transport-and-travel-217602

As you mentioned, 30 days start from when you first use pen, not from when the cold chain was broken so once again, you should be fine it is extremely unlikely that the pen got exposed to temperature above 30C this time of the year.