I think Scotland raised the age boundaries fairly early on so that people (the NHS website says women, I'm guessing by that they mean people with cervixes) up to the age of 26 (I think) could ask for it (and it would be the standard prescription cost per jab). When I got it done they asked if I had had any sexual contact with anyone, but as I answered 'no' I don't know if it would have been a problem if I answered 'yes' if that is at all relevant to your situation (I got it done at age 19, two years ago, I am almost dead on a year younger than you :) If I was born on my due date it would have been exactly).

I'm trying to google to see if England and Wales also raised the age boundaries, but all I can find is that it is not "routinely" offered to those over 18 years old. At the very least it's worth asking, it certainly can't cause any harm. (The Scottish NHS is very separate from the English and Welsh NHS, so what happens in Scotland doesn't necessarily reflect the situation south of the border)