Sunnyg, I retired from MN six months ago but still lurk. When I saw your post I joined again under a different name so I could let you know of my experiences
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We used to live in Islington and started our infertility investigations at the time. The GP did basic day 3 and day 21 tests and sperm analysis via the local hospital and then referred us to UCH for further investigations as DH's sperm were slow (not low, just lazy!)
I got the policy on fertility treatment from the PCT at the time (2008) and it stated that on referral from the GP, UCH will carry out investigations and, if appropriate and necessary, offer up to three IUIs. If these didn't work then UCH would refer the couple to IVF Hammersmith for up to two fresh cycles (IVF OR ICSI).
PCTs buy fertility contracts with hospitals/clinics and it is not always the nearest one and can be split (i.e. when we did it investigations and IUI were done at UCH and IVF was done at the Hammersmith Hospital!). Unless you have seen it on the policy or actually spoken to the PCT do not assume that if the investigations and IUI happened at UCH then IVF, if you need it, would happen there too. The IVF clinic at UCH is actually a private one called Centre for Reproductive and Genetic Health (CRGH). I am currently there as a private patient and can certainly recommend them; they have some of the best embryologists in the business and a customised approach to IVF dependant on your personal diagnosis.
As for your timing in terms of the move, I would organise the re-direction of mail and hold off registering with a new GP until your investigations/treatment at UCH are well underway. I know this is naughty but, the mere fact that you have got the appointment means that the PCT has already committed the money for you and your initial treatment (investigations and IUI). If these do not deliver your much wanted baby and the consultant starts talking about IVF (we had a meeting with her between IUI 2 and 3) then say that you have recently moved and are under a new GP and PCT hence the criteria may change. Get a copy of all your notes so that when you register with your new GP and he starts mentioning investigations you can prove you have had them all done and are ready for the IVF referral straight away.
One of the things I regret not doing during my NHS funded IVF cycles was to try to do a private one whilst I waited. It depends what the PCT policy is, some will strike you off the list you did a self funded cycle whilst on the waiting list, others do not mind. Start saving now if you possibly can and if you do face a need for IVF but a year long waiting list for the Homerton then you could do a private one at a clinic of your choice whilst waiting if your PCT lets you.
I have only had experience of UCH for IUI and ICSI at the Hammersmith and CRGH so I am unable to comment on how good or bad the Homerton is. There is a great website called Fertility Friends where the posting boards are clinic specific so you could visit and read up what people say about it. Remember that if you are on an NHS cycle you cannot normally choose a clinic or timescale and the PCT/clinic with the contract will dictate everything and you have to go with their flow.
Sorry for the essay, assisted conception is such an overwhelming subject that I know I was very grateful at the time for anyone with information on it. If you would prefer to PM me with any questions please do 
Good luck, it is a real adventure but has resulted in so many dreams coming true for so many people that it is certainly worth it!!!!