Yeah with a spinner you will only be able to compete at events run by Robo Challenge as theirs is the only arena that can safely contain spinners at the moment. With an axe you will be able to run at Roaming Robots and Robots Live events too.

The link you've provided is only for the Planet 5 transmitter. You would need to buy the receiver to go with it, which will probably cost ~£15 so that'd be right up at your total of £50.
Alternatively, there is the 2.4GHz 4-channel transmitter and receiver combo from Giant Cod RC for ~£26:
http://www.giantcod.co.uk/gianitcod-24g ... 03779.html

That is a basic-function transmitter/receiver, it will allow you to control your robot and has inbuilt mixing if you wish to have one-stick control, but features such as dual rates etc (commonly found on more expensive transmitters) aren't present but also aren't essential to the control of your robot so it would be a suitable alternative to the Planet 5 set.