What do you mean with "my medical states only the British details"?
Your medical is issued, linked against your license. Not against your passport or other certificate of citizenship. The only (weak) link between license and ID is that your medical and license do not have your picture on it, so you need to be able to show some sort of official picture ID if anybody ever ramp checks you and wants to verify that the license really belongs to you. But that picture ID does not need to be of the country that issued you your license.
Here's what you need:
- A license/medical combination that allows you to fly the aircraft you intend to fly. For an EASA-registered aircraft (G-reg, D-reg, PH-reg, whatever) this means the old-style JAR-FCL license or an EASA license. For an N-reg you'd need an FAA license, and so forth, although there are exceptions to this.
- Your license needs to be at least at the ICAO PPL level to fly internationally (otherwise the Chicago treaty does not apply, or something along those lines). For flights within the EU, you need at least an EASA LAPL. The old, national NPPL, RPL and similar licenses in principle do not allow for international flight, although countries have set up waivers for that in the past.
- You need to have the proper passport/identity card/visa to leave the UK and enter the Schengen area, and vice versa. As you hold both German and British citizenships, and presumably the means (passport or citizenship card) to prove that, it's not going to be a problem at all.
Do remember that entering or leaving the Schengen area can only be done at a designated customs/immigration airport. You cannot just enter or leave the Schengen area at any airfield you fancy - and note that some airfields no not have customs/immigration 24/7, or require prior notice. At the UK side things are slightly easier, because of the GAR form and surrounding procedures.