Hi MD, good to meet you.
London is surrounded by flying clubs, and it helps to know what part of London you'll be in to make recommendations - but I can promise there'll be options.
You can get a briefing by an instructor (we call then FIs, "CFI" means Chief Flying Instructor here) and an examiner sign off that you understand how to fly in the UK - that will do you for a year. After that you need a UK licence. However if you have an FAA PPL and over 100hrs, you can do a quick conversion, quoting the current guidance material (Part Q Subpart 2 of CAP804, technically obsolete and pre-Brexit but still widely used as guidance material for issue of UK licences and largely still valid)...
B. CONVERSION OF LICENCES to Part-FCL Private Pilots Licence
(1) A PPL/BPL/SPL, a CPL or an ATPL licence issued in compliance with the
requirements of Annex 1 to the Chicago Convention by a third country may be
converted into a Part-FCL PPL/BPL/SPL with a single-pilot class or type rating
by the competent authority of a Member State.
The pilot shall apply to the competent authority of the Member State where he/
she resides or is established.
(2) The holder of the licence shall comply with the following minimum
requirements, for the relevant aircraft category:
(a) pass a written examination in Air Law and Human Performance;
(b) pass the PPL, BPL or SPL skill test, as relevant, in accordance with Part-FCL;
(c) fulfil the requirements for the issue of the relevant class or type rating, in
accordance with Subpart H;
(d) hold at least a Class 2 medical certificate, issued in accordance with Part-
Medical;
(e) demonstrate that he/she has acquired language proficiency in accordance with
FCL.055;
(f) have completed at least 100 hours of flight time as a pilot.
Realistically, you'll need a day or two on the ground and half a dozen hours flying training to come up to speed on all of this, but my advice would be to find a good flying school - there are many - let them sort the paperwork out and just dive in and do it.
The UK skill test is like the flying component of the US checkride. The written exams here are tougher than the FAA versions, but you only have to pass two, and the study material is widely available.
For the IR, if you have a current (current really matters here, which depending upon the interpretation this month may mean you've passed an IPC in the last 12 months - my recommendation would be that you play it safe by taking an IPC with a local DPE just before you come over), and at-least 50hrs PiC IFR then again you can just book a skill test (again, basically the same as the airborne part of a US checkride, although they may elect for a conversion from FAA to conduct an oral portion also: this is at the discretion of the examiner). My recommendation again would be to plan on half a dozen hours instruction with an IRI (Instrument Rating Instructor, equivalent to a US CFII) before the test - things aren't done identically here to the USA and you will need that.
And then you'll have a UK PPL/IR, and the same rights and privileges as everybody else. There are loads of opportunities to rent or buy shares in G-reg aeroplanes in the UK, N-reg aeroplanes exist, but they're largely privately owned by their sole operators, and both the UK and EU are in the process of clamping down on these operations, so frankly it's probably easier just to stick with the G-reg route.
A couple of notes - whilst the FAA are only interested in time by sole reference to instruments (under the hood or in cloud), here there is a distinction between sole reference time, and IFR time (even if the latter is in blazing sunshine, clear skies, and no hood). For eligibility to take the IR skill test, it's time PiC IFR that counts ONLY. For the record, I have both UK and FAA IRs (and CPLs) so have been through all this myself. Secondly, in the UK the "night qualification" or NQ (more often referred to by its old name of "night rating") is a separate add-on to the PPL. Frankly, don't bother, it's not worth it - night airfield lighting is so rare at UK airports you'll never use it. IFR capability is useful in the UK, night really isn't: if you do want it, it's another 5 hours of training, I don't *think* there's a dispensation for that, you just have to do it.
Currency requirements are different here to FAAland also. You're looking at a biennial "flight with an instructor" similar but less onerous to the US biennial flight review for the PPL class rating (which will be SEP, or Single Engine Piston, broadly equivalent to the US SEL), but you are also looking at an annual IPC to keep your UK IR current, we don't have anything equivalent to the FAA's rolling "6 in 6" IFR currency, it's just an annual flight test.
You may hear of a simpler UK rating called the IMC rating or Instrument Rating (Restricted). I can't think of any good reason for
you as an FAA IR holder to go that route, it is less capable (UK airspace only, no class A), and still requires periodic (2 yearly) retesting. I'd go for the full IR, because you can.
Best of luck with the trip over - whilst not as user friendly as much of the US, and more expensive, the UK is nonetheless a great place to fly light aircraft, with huge numbers of choices, clubs and airfields. With a G-reg aeroplane, UK PPL/IR you can go anywhere in Europe from here.
G