It used to be, walk in to the FSDO...
Now you need an appointment, or not, where in certain parts of the US that they delegate this duty to a DPE, who charges a $200 fee for the privilege!
The process also now requires a license verification letter from your NAA.
Once you receive your 61.75 certificate, you need a current Flight Review (see Collins, above) and medical (FAA or foreign will do) to act as PIC. FAA recency requirements also apply to carry passengers.
Despite it being “Foreign Based” certificate, it is a real license - US privileges apply. As long as the foreign license has not been “surrendered, suspended, revoked or expired”. This means that even if you are not 'EASA' current, or typed (eg PA-46, Single Engine Turbine, etc), you can still be legal to fly on FAA/N privileges. (There’s another opinion letter on this, still looking...)
If you have Instrument privileges on your foreign license, these can be added to the FAA certificate, upon passing an ‘Instrument for Foreign Pilots’ exam ($160). Complete this before applying for the 61.75 certificate, or arrange a second appointment or another $200 to a DPE.
When EASA licenses were introduced, all the license numbers changed. This immediately rendered all those 61.75 certificates invalid. For a short period, now expired, the FAA had an express re-issue program. Now you need another appointment.
With Brexit, new license numbers may feature again. And if you switched NAA, you’re out of luck. Only one State's foreign license can be used, ever [61.75(f)].
As for EASA aircraft, the jury is still out on that question. With an (ICAO compliant) UK national license (not for EASA aircraft types), can you fly an N-registered ‘EASA’ aircraft? The opinions in the previous post seem to suggest yes, but - I’m not a lawyer!
With regard to Type Ratings, any held on the foreign license will be transferred, or can later be transferred. In practise, these are worthless. For both the TSA & a major Part 142 school, they were not recognised as holding the qualification to permit an immediate check-ride. An approved course would be required.
A final word on the foreign-based certificate: if you first gain an original FAA certificate, you can no longer apply for a 61.75 certificate [61.75(b)(3)]. But you can do this the other way around. [For example: let’s say you complete the requirements for an initial issue of ASES (Seaplane), if you also hold SEL/MEL/IR on a foreign certificate, you can no longer apply for the foreign-based certificate. If however you apply initially for the foreign-based, and then gain an independent certificate, you may still update the foreign-based certificate and use those privileges in the US, or on N-registered aircraft.]
There is an option to add privileges as “US Test Passed” on a 61.75 certificate, but since this is only at the Private level, there is no real advantage.
If you were previously issued a Commercial (Foreign Based), it is normally re-issued at that level. But with the 'no compensation or hire' restriction, it is essentially a Private certificate.
As for answering your question, I'm not qualified to pass judgment, I find it hard enough to keep up with my own privileges! Hopefully there’s enough reference above to make an informed decision.
Good luck.