Sadly not nearly as simple as going to the Post Office and buying a license.
Our rivers, lakes, canal and harbours are owned and run by a variety of companies and landowners. As has been suggested, if you wish to paddle a canoe or kayak then joining the BCU will get you a waterways license which will allow you on British Waterways, Environment Agency and a few others waters. You also get a third party insurance policy up to ten million pounds, insurance is a requirement for many other waters like reservoirs etc.
If you wish to use inflatable or other types of dinghies then the license for unpowered portable boats from British Waterways is about £40. I'm sure the EA have a similar license, but I've not looked into that.
A quick google gives this URL for Staunton Harold Reservoir:
http://www.derbyshireuk.net/staunton_harold_reservoir.html
It is unfortunate that many stretches of water are run by private organisations and companies and not covered by the BCU agreements. I tend to pay any fees to help cover costs of what ever groups owns the water to maintain any facilities I use. My choice of course, but there you go.
Al