Most local councils now have a register of "approved" tradesmen ~ in many areas this includes a check so that users may feel safe that the tradesman they select from the list has been vetted to work with vulnerable adults and/or children but if you want to go down the tutoring route there is NO legal requirement to be CRB checked. You must, of course, act in accordance with the "Protection of Children" Acts.
We ran a tutoring company for a number of years when we first moved here. We stopped because of the CRB checks costing so much. As we employed the tutors we had to have them CRB checked BUT an individual tutor does not need to get clearance. That is up to the individual parents. Wife still does tutoring but as she also works in the public sector she has been CRB checked but she has no proof of that should anyone ask to see it. Nobody who has placed their child with a tutor via our business or with Mrs S#2 has ever asked to see any evidence of the checks having been done. Clear proof, to my mind, that it is another piece of worthless red-tape to raise revenue for a bankrupt government.