I am an instrumental music teacher who also runs pre-school music classes on an independent basis - I've been doing it forever (or so it seems) so I can probably shed some light on some of your queries.
Firstly, re tax - at this stage it's most logical to set yourself up as a sole trader and then register with the tax office as self employed. You need to do this within 3 months of beginning trading or you could be liable to a £100 fine. If you don't earn anything, you don't pay any tax. You need to keep sensible records of incomings (ie class fees) and outgoings such as hall hire, insurance etc. Keep all your receipts, for absolutely everything. Set up a bank account for your business and keep this separate from you personal banking - there are some which are free for small businesses.
Make sure you get public liability insurance - Michael Morton or Blackfriars are good starting points or you could try the Pre-School Learning Alliance.
CRB checks - not necessary but would assume as a primary teacher you already have one.
Regarding using pre-recorded music in your sessions... well, it's a little bit of a minefield. You can get a Public Performance Licence (PPL) which would cover you for people who have registered with them, but a lot of pre-school music isn't covered by this. Some people require you to take out an annual licence to use their music, others may grant you free use if you sell their products. I now produce and use all my own music so can (thankfully) avoid this issue. Best idea is to contact the producer of any CDs or downloads and make sure first.
You are freely able to use any music in the public domain such as nursery rhymes or action songs if you sing them yourself unaccompanied or provide your own backing music, either pre-recorded by yourself or if you play guitar or keyboards for example.
Making a living from it is perfectly possible but do allow yourself around 18 months to two years before you make any reasonable profit from your efforts. I've been running my classes for around 8 years and have a good 5 figure turnover, but would lose around a third of that in costs such as hall hire, insurance, transport, equipment etc. It takes time to build up a market presence and even then you have fluctuations in how 'popular' your group is depending on which uber-mum is attending whose sessions - it's very word of mouth driven and sometimes one person will send everyone in one direction, which is great if it's yours... not so good if it's not!
Anyway, before I send you off to sleep - it's fab job, but bloody hard work. You need to spend time, money and effort to get it off the ground and don't expect instant results. Give yourself time to grow into the role and develop a good product.
Best of luck and if you have any specific queries do ask.