House price changes is a valid point. House prices in the UK have fallen generally in the last year except, notably or the sake of this discussion, in most of London and the immediate vicinity so you really need to check the particular area carefully in respect to this. Prices in Crawley for example have risen but probably not in all areas of Crawley.
The advice to rent for the first six months is an excellent one. There is no shortage of rental property and although not cheap (the high house prices of London and environs makes sure of that) the time renting allows you to investigate something more permanent could save you an expensive error.
I also agree that you should budget £2,000-£2,500 for legal and other fees and perhaps another £500-600 for a more detailed survey on the house. There are several levels of survey: a basic one to satisfy the lender that the house is a good credit risk (this is NOT recommended as the only survey conducted), a medium level one that checks most things but does not include investigation into structural matters unless they are visibly obvious, and a full structural survey where the surveyor pokes into all aspects of the house and it's structure, pulling up floorboards, digging down to foundations etc if necessary. Needless to say, the cost increases accordingly. The seller pays all the Estate Agent's fees so that will not effect you as a buyer.
House prices can be difficult to understand as an otherwise identical house 100 metres away can be worth substantially more due to a number of reasons such as school catchment areas, proximity to transport links, distance from less desirable areas etc. This is heightened in and around London due to the high population density. Because of this, I would not advise buying a house on the back of a few visits to the country; wait until you have been here a while and can view the local area at your leisure and at different times of the day.
A useful site to investigate a particular area, once you have narrowed your favourite areas down a little is
www.upmystreet.com. You type in the area - preferably the full postcode (it will take partial postcodes, whole town names or named areas of towns and cities but a full postcode is much more precise as it identifies an area of a handful of houses). The menu on the left hand side then allows you to investigate that area.
I've lived in West London most of my life so am not really closely familiar with Crawley at all (except that I know it is a popular commuter town) but a general rule is that there are nice, and not so nice, parts of any town and they can often be within 200 metres of each other - hence the need to investigate on the ground, in person.
To recap: do rent for the first 6 months while you sort out something more permanent. Your costs over and above the house will be £2,000-3,000 + 1% of the house price up to £250,000 and 3% of the price over £250,000 (up to £500,000 when it becomes 4%) plus the cost of moving your goods to England. Council taxes are typically £1,200-1,400 per year.
Oh, and you will not be alone in London being from Italy: it is estimated (exact figures are impossible) that 1/3rd of people currently living in London were born outside the UK and there is of course a large Italian community here.
Good luck, Bob.