Not quite that simple. FSDO's are not necessarily on your doorstep (200 miles in my case) and FAA will need confirmation from CAA that your license is valid (it could have been revoked!).
Generally speaking there is a bit of variability between FSDO's and the people in Oklahoma City, sometimes the rules say one thing and the practice is different. In my case I was lucky and was able to meet the FAA person 30 miles away where he was getting ready to supervise a big airshow so it saved me a 400 mile round trip.
The "activation" bit is something of a red herring. The FBO will want it but opinion is divided as to whether or not it's a legal requirement. The enlightened say this is the FAA way of "rendering valid" a foreign license so if your UK license and medical are valid and you would be legal in the UK then the 61.75 makes you legal in the US. Others point to FAR's which conveniently require holders of certificates to have BFR's and medicals without differentiating between a standalone US Airman's certificate and a 61.75.
You'll find this useful
Airmen Certification - Verify the Authenticity of a Foreign License, Rating, or Medical Certification