italianjon, this is a legal question so it needs to approached in a legal way.
You are flying in a German registered airplane in German airspace with a UK license. We can forget about the license here, it is not relevant to something like aircraft equipment. Oxygen is not about preventing license holders to climb higher than altitude X but about requiring the use of supplementary oxygen or a pressurized cabin.
Airspace could be a factor, there is law requiring you to carry certain equipment while flying in certain airspace. For Germany one example is the DME for every IFR flight (a lot of N-regs don't do this and don't know that they can be fined with up to 50 000 € for that). In this case, there isn't anything related to oxygen.
What remains is the law pertaining the equipment of aircraft. More specifically, there is § 11 of the 3rd implementation ordinance of the regulations for operating aircraft,
3. DV LuftBO in German legal speak. The LuftBO applies to all aircraft on the German register and its
§ 21 contains the requirements for oxygen but limited to commercial flights. The 3. DV LuftBO adds requirements that ally apply to non commercial flights. Here is what it says regarding oxygen:
(1) The oxygen required to be carried on board depends on the altitude, duration at certain altitudes and the number of persons on board.
(2) Flights with cabin altitudes of more than 3600 meters (12 000 feet) may only be conducted if the oxygen carried on board is sufficient for the following:
1. Supply of all crew members as well as 10% of the passengers if the flight duration at a cabin altitude of 3600 meters (12 000 feet) exceeds 30 minutes and
2. Supply of all crew members as well passengers for the entire time at cabin altitudes of more than 4000 meters (13 000 ft).
(3) The PIC shall ensure that the required supply of oxygen is available on board.
...
The provision applies to all aircraft, not only commercial. Note that it only talks about "supply available", it doesn't require you to actually breathe oxygen.
Unless you're in a C152, 30 minutes is plenty of time to cross the Alps although 12 000 ft is not enough in all areas. The US rules on oxygen are more relaxed than what we find in Europe.