If you care regularly for somebody else's child for more than 2 hours a day than you need to be registered. You don't need to be registered if no money changes hands, so reciprocal arrangements are fine.
Registering is a big hassle and expense, there's no doubt about it. You need to pay for Ofsted registration, public liability insurance etc. and will be inspected against how Ofsted think you should be doing things. You'll need to undertake first aid and child protection training. You must do a training course before registration (mine was four full days plus about 12 assignments).
If you are caring for a child under 5 (until end of reception year), you will also be expected to deliver the Early Years Foundation Stage Curriculum. I don't have a huge problem with this curriculum - it is play-based and totally child-centred - but the planning and record keeping involved is fairly hefty.
All parents using registered childminders can claim the childcare element of tax credits (depending on family income). You just need to give the childminder's registration number to the tax credits people.
purplypink is right, there are different rules for nannies. That might be something worth considering.
I don't think that providing home-education (rather than 'just' play-based care) would be a problem if you were taking the autonomous approach. You could argue that the child's education was still entirely the responsibility of the parents (so long as you weren't making the child go through endless workbooks/formal learning). Providing general play-based educational activities is something that all childminders must do anyway.
Do think about your options carefully. As an already registered childminder, I would love to care for home-educated children, but I don't know that I would have gone through the whole registration process just so that I could care for one child...