I've travelled to and from the UK with an Acoustic guitar, an Electric guitar and my mandolin on different trips. No more than 2 instruments at a time and have had no problems. My cases have suffered some surface wear and tear, but the instrument has been fine. I've check the guitars into the hold and made sure to loosen the strings first. Did have to get an adjustment on the truss rod on one trip over with the acoustic. The mandolin I put in the overhead compartment.
I would suggest a really nice case for your guitar, do a google search for flight cases, if it is a more form fitted case you will have a lower chance of it getting tagged for oversized bagage fees.
If you are bringing one insturment this would be my suggestion. As for Amps if you have some special hand built boutique amp ship it over and get a power converter. If you have some mass produced something or other, sell it and buy a new one in the UK. The power is different. When comparing the same model amp for US and for UK you will notice a difference.
I would like to hear some input on shipping companies to get my stuff over there.
I plan on moving in January, and have about 6 guitars, 3 mandolins, and a bass. I would spend 3 or 4 times the money to replace these instruments in the UK because you just don't find quality instruments at the kind of deals I was able to obtain mine. Plus I'll have a couple boxes of other stuff like clothes. books, and leatherwork tools.