It is fine for travelling to use seat from out of country. It's just if you were moving there you would have to get a European seat.
This is correct. I would however be aware of a few of things if using London taxis (I make the assumption you are referring to "black cabs" so do correct me if this is not what you mean!!)
In general, cabbies are famous for their london spirit and chit-chat and are very friendly. Equally, however they are interested in efficiency and profit and although they
should allow it, don't bank on actually being able to find the time to install the seat correctly. They will most likely want to get moving and the time you spend installing the seat, you
will pay for.
Second black cabs are a VERY expensive form of transport and if you are concerned about getting to a hotel from the airport, be it Heathrow, Gatwick or London City, I strongly recommend using the tube or the DLR. It will be
much cheaper.
Thirdly be aware, you may require a locking clip as European vehicles are rarely fitted with locking retracting seatbelts. London Taxi's I'm sure are not.
I don't mean to put you off the idea of using one, they are very good, just be aware it might be a bit rushed and a little more complicated than you may have anticipated as well as costly.
Using the seat in other vehicles is the same as anywhere I guess, again making provision for possible use of a locking clip.
If you need to do travel planning for London, I suggest visiting
www.tfl.gov.uk. Also see
this post from a while ago. It still very much applies!
I hope this helps and enjoy London!