I had mine done on the NHS at 25, ten years ago. It's on a case by case basis. Generally they won't if you're under 30 but I was allowed after convincing the surgeon...
First was an emergency section so the second time they give you the choice. I had my pre-op for the planned section and I asked while I was there. I said I'd thought about it and stated my case. He asked me lots of questions like was I married, do I want more children, what if we break up, what does my partner think (he was there so he was asked as well). He asked what would happen if they didn't do it. I'd already prepared my answers after advice from my midwife. I declined counselling which is the usual process, and continued to give my reasons below...
I have a genetic illness.
It was my second c section and they dont like doing more than 2 because of the risk and cost to the NHS. After 2, you're almost definitely going to have to have a section over a natural birth.
I'd had pre-eclampsia, gestational diabetes, hyperemesis... just difficult pregnancies that were both back to back due to my womb.
Can't take hormonal contraception because of migraines at the time (which have resulted in proximal DVTs in later life).
I'd actually been pregnant 4 times and due to the two that were lost, I almost died from hemmorhages and had to have emergency surgery.
First labour was 5 days before emergency c section and due to no sleep in that time I got PPP. Then got PPD, and I'd been warned that could happen again.
... well the surgeon said that he would sterilise anyone presenting with just one of those reasons and he put it on the same surgery. Just mentioning in case anyone is looking in future.