a resource for parents and students
This is a much harder question than either keyboard or guitar. Of course, if you already have a piano, you are laughing and probably wouldn't be reading this page anyway! But if you don't, and you would like your child to learn piano, it can be a very expensive decision.
Many people think that it is all right to have a keyboard at home on which to practise. Sadly, this is not really the case and I wouldn't recommend it. The technique between the two instruments is quite different. To be fair, some of the very advanced keyboards may be more suitable than others. But you would be looking for a minimum of six octaves (most keyboards have five and a piano has seven). You also need one with a very good touch sensitive keyboard and of course you would need a pedal attachment.
There is another option, of course, and that is to rent a piano for a specified time while you see how your child gets on. It is a long term commitment, however, and not something that you can decide in a few weeks. Some places will do a 'rent-to-buy' scheme. The advantage of this is that the maintenance of the instrument is part of the package. There are several places in London that rent out pianos, but I cannot recommend any as I have never used them myself. But a simple search on the internet is a good start.