Unfortunately I don’t think you’ll be able to complete both at the same time in the UK.
Like the FAA commercial, you’ll have to do a written test ( actually 2 unless your exempt the FOI ) which can no longer be done in the UK. No endorsement needed from CFI beforehand to enable you to sit the tests.
The other problem is getting an examiner for the FAA test in the UK. Until recently, for an initial FAA instructor certificate the “local FSDO” had to allocate a DPE although that rule has recently been relaxed but the availability of a DPE in Europe / UK has been covered many times previously on pPrune.
A suitable qualified FAA CFI based in the UK should be able to give the required training / endorsements / iacra that are needed ( refer to part 61 for the required ground / flight training that is needed ) and fortunately you won’t need to undergo TSA screening prior to beginning training.
https://www.ecfr.gov/cgi-bin/text-idx?c=ecfr&sid=40760189a03dfea0b501608f33820a45&rgn=div5&vie w=text&node=14:2.0.1.1.2&idno=14
There used to be ( and there may still be ) EASA approved flight schools in the US that did EASA FI training and if you wish to do both in the same place / at the same time, then the US may be a better / only option.
My own personal recommendation to you, would be to do them separately, the EASA FI in the UK at on-track aviation and the FAA CFI in the US at the school where you did your FAA commercial.
The FAA are still using the PTS for the CFI and along with the endorsing CFI will both give you an idea on what to expect in the practical test.
https://www.faa.gov/training_testing...-S-8081-6D.pdf.
VFR ( EASA FI / FAA CFII )