Jan
There are no ''Rules'' for oxygen use for UK registered private flights, only sensible recommendations.
Regulations apply to Commercial Air Transport operations with fee paying customers.
Flight for short periods ie crossing the alps above 10000ft. should pose no problems for a fit person.
I believe EASA of course, are trying to regulate private flights as AOC ops in the future.
(one of the initial requirements was a question for rate of climb after engine failure for single engine aircraft!!)