Technically you cannot 'convert' anything. Although prior training and experience is accepted.
First you'll need a license verification from both the CAA & FAA.
You can sit the optional Knowledge Test “Instrument for a foreign pilots”.
Your new (foreign based) FAA certificate will give you private privileges, (and IFR).
Before you solo/PIC, you'll need a Flight Review (and IPC).
These would typically be completed as part of a checkout process anyway.
The next stage will be training, as required, to meet the ACS standards for the Multi & Instrument check-ride.
Before your check, you'll take the 2 Knowledge Tests, and verify you meet the aeronautical experience required.
If you want SEL privileges as well, you'll need another check which includes some manoeuvres you may not be familiar with. If you have the hours/competence for the EASA CPL/IR, it shouldn't take too long to complete.
I’d recommend a Part 61 school, rather than Part 141, as the latter has to fly according to the syllabus.
You'll also need TSA approval for both the Multi & Instrument. But since you already have the “foreign qualification” this can sometimes be conducted on a 'B' Visa, which gives you flexibility to change schools.
I realise this is an old thread, but since the question comes up regularly, this is an accurate summary.
Good luck and enjoy.