Medical is changing...
To the OP:
Eyes are a completely waiverable item. I have a "Statement of Demonstrated Ability" for vision. That was a one-time thing. I also have an AASI - AME Assisted Special Issuance - Aviation Medical Examiner Assisted... Abbreviations w/in abbreviations. The AASI is only valid for 12 months. The paperwork is a bit onerous, but has been worth it. I've just worked out that all my required doctor visits are scheduled in the several weeks leading up the medical expiring. The docs write the same letter(s) every year.
About the meds, you should be honest, because if you are not, then there are all sorts of problems if anything comes out.
Is your FAA certificate a stand-alone certificate or is it based on your British one? If it's the later, then it needs to be valid, too.
When are you planning your trip? It should not be much longer until private flying in the US will take place without a medical at all. The bill was passed last summer, and congress gave the FAA 12 months to develop the implementing rules or it would take effect on its own.
Also, even w/o a medical, you can become current and fly around with an instructor. For an American logbook that's still PIC because you have an ASEL (I presume) rating. EASA would see it as something else.