Yes............ the "authorised instructor" bit has been done to death and this is what is used by some money-grabbing UK based FAA training operations to say to candidates that they must fly a minimum 15hr package with them.
This was very much in vogue during 2004/2005 but I have lately noticed that most no longer run this little scam. There is one well known chappie who still does, but then he lives in his own universe which only occassionally intersects ours
The instructor can be any instructor anywhere in the world who is authorised to do the training which he did.
This is how you get your previous UK- or whatever-based training admitted towards the FAA license/rating. But as I said above, there needs to be no ambiguity as to what each flight was - for example if you claim to have done a 250nm x/c then it needs to be between two airports which can be obviously shown to be at least 250nm apart.
As to what training is required to reach the checkride standard... this will obviously vary. Prior to me doing the IR in the USA, I had IIRC about 70 hours IFR (logged as actual IMC with the autopilot disengaged) which included about 25 hours from the IMCR training. I had also been doing airways flights (as PU/T) around Europe and was able to fly any SID, STAR, IAP, enroute, do all the flight planning, etc. so for practical purposes I already knew what an IR holder would have actually needed to know.
However I still needed most of two weeks i.e. about 25 hours' hard flying under the hood, to reach the FAA checkride standard. This was the hardest flying by far I had ever done in my life. The hood goes on seconds after takeoff and you spend the whole flight doing vectors, then banging VOR/DME/LOC/ILS approaches, every one to minima, and you don't know if you will land or G/A until you are AT the minima and then the instructor lets you know. One would be a fool to listen to the scores of idiots who say the FAA IR comes on the back of a fag packet and is no harder than the UK IMCR. It is MUCH MUCH harder.
Nevertheless, a competent pilot who already knows how to do it all should be able to do it in 2 weeks, or 3 weeks if training on a modern G1000 equipped plane.
Due to the difficulty in the flying I would always recommend people to get all their ducks in line (exams, etc) before they commit to either going to the USA or booking an examiner in the UK.
The visiting examiner I know charges $600 per checkride, plus a share of expenses. This is IMHO very reasonable, looking at how much money and aggro you save not going to the USA, and the benefit of doing it all in your own plane in incalculable. If you can't afford this, you won't be doing any IFR flying anyway.