I have signed many athletes from the UK out of their equivalent to our HS and several after they attended 1 year at a UK University. With that being said, you might be surprised at how different one USA University is compared to another in terms of competitiveness.
I would look at the Universities top 2 athletes in your events (400/800/1500 ) and if you are running times similar or better reach out to that event coach and there maybe possible scholarship dollars for you. Also, look at those athletes classifications (Fr, So, Jr or Sr) as you may be talented to run for the University but if they are predominately 1st or 2nd year athletes there maybe no scholarship money available (USA Universities have a set scholarship allotment for T&F and it various from University to University)
In terms of eligibility, the athletes I had who ran in the BUCS races lost 1 year eligibility for that sport. If they ran for their club they did not lose a year. It kinda sucked because most of the Uni teams had no coach and it was a bunch of athletes who threw on a Uni singlet just to run 1 or maybe 2 races to represent their Uni in the BUCS races.
Good luck