Every pet entering Great Britain needs an ISO 11784/11785-compliant microchip, and the order matters: it has to be in place before, or at the same time as, the rabies vaccination. A vaccine given before the chip doesn't count, and has to be repeated once the chip is implanted, which restarts the clock.
The rabies vaccine itself can't be given until the animal is at least 12 weeks old, since that's the minimum age the vaccine is licensed and effective for. If this is the pet's first ever rabies vaccination — or a previous one has lapsed — you then have to wait at least 21 days after that vaccination before travelling. A pet with an unbroken, up-to-date vaccination history doesn't need to re-serve the 21 days for a booster; the wait is specifically for a primary course.
Ask your vet to scan and record the chip number at every appointment — vaccination, the AHC visit, and the tapeworm treatment if applicable. A mismatch between the number on the chip and the number on the paperwork is a genuinely common reason for pets to be held up at a GB port or airport.