And please don't have your child tutored for the Maths Challenges - that is totally not in the spirit of the event. There are plenty of resources on the UKMT website in the form of past papers etc.
As for secondary schools, HBS being a superselective, you have until the day your daughter starts there to move to somewhere more convenient than Croydon, so there is no penalty involved - other than the price differential between North and South London, the add-on costs of a house move etc -. in her having a go at the entrance exam (remember that it is in two parts and only a fairly low percentage of those sitting the first part will be deemed suitable to sit the second).
But if your daughter is already in year 5, I would look very carefully at the residence requirements for any selective school which does not admit purely in score order, as you may have to have moved before October 31st (final date for submission of your CAF) - or even by the time of registering her for the exam - in order to be considered in catchment and therefore have a better chance of obtaining a place. You can, of course, take a gamble and apply for several schools as out of catchment (assuming that your daughter will have done well enough in the relevant entrance exams for this to be worth contemplating) and plan to move if she gets in somewhere, but remember to include a local school for which your DD will be eligible, should your more aspirational preferences fall through.
If you are going to enter your daughter for 11+ exams, registration is already open at many, if not most, schools so be careful not to miss the closing dates.