Hi frogg6- I shall try to give you as much info as possible based on my own experience and what I know of ways to get there.

Depending what you want to do in an undergrad degree, you can then go and do a masters or straight into work.
If you do a physio (hons) grad degree, then I'd follow the advice of everyone else and try as get as much work experience during your free time OTHER THAN GIVEN PLACEMENTS AS PROVIDED ON YOUR COURSE, in the NHS, during your study.
I'd look into as many sports-specialist courses as possible, and even try local (non-professional) teams for voluntary work.

You can also choose then if you want to do a sports physio/therapy masters.
Same goes if you do a sports science/medicine undergrad, then do a physio/sp physio/ sp therapy masters.

I did a Sports Therapy (hons) degree and whilst at uni I voluntarily worked with clubs at uni doing pitch-side first aid, worked at a local physio practice, a local sports injury clinic, then progressed in 2nd yr to a local football club, by 3rd year i was working with 2 local football clubs, uni teams, local rugby club, basketball and american football club.
It's good to note that unlike physio degrees, you generally have to go and find your own experience.

Post uni, I yet again worked in more clubs and clinics, mainly voluntarily, worked in two football academies, with a premiership reserve team, and now work as a head therapist in the FA, work with a mens and womens football club, rugby union club, in my own practice, and within a national sports institute. Hard work and experience will pay off!

Oh and if you want to work in sport, make sure you do a sports massage course, and do a pitch-side/sports first aid course as otherwise you're not legitimately covered to work in clubs.