Help?!

Hello, I’m an American living in the UK with my UK native fiancé, I’m expecting but on a visitor visa… Planning to update to family visa when I can. I’m just curious if my baby is born over here with a British father does that make it a UK citizen by default? I don’t want to do the whole birth thing in the states because of the cost of visas that and it’s ridiculously expensive ($40 charge for skin contact with the thing that I grew and pushed out of me? No way)