Sadly (for us pilots) EASA is not a "country" nor is it an ICAO signatory. The individual members of EASA are countries and issue EASA compliant licences that they have all agreed to honour but the USA is not part of the club.
The bilateral agreement has been long promised but not yet happened.
The CAA will issue a permit on payment of a fee (I think it was £75 last time I did it), production of certain documents and a reason to be flying the aircraft here. A month should not be a problem but each new application will have to be paid for again, so think it out before you apply. They are efficient and work via email and phone.
Each country that you wish to fly in, apart from the UK, will also need a permit, as well as any that you need to overfly to get there. The procedures to get these are different for each country and can be quite expensive and/or restrictve.