Err...
Not quite.
Bristow Helicopters have certainly trained some people from scratch (although I believe most of their crews still have a military background). I would guess, but cannot confirm, that CHC and Bond, who with Bristow are the companies currently providing civil SAR assets in the British Isles, also have some training mechanism in place.
In all three cases their requirement for SAR rearcrew is fairly small, so I guess they have a course and trg staff that gets activated as required.
The UK military SAR trg is provided at RAF Valley by a unit called SARTU, which uses Bell 412 aircraft, and some crews, provided by a contractor called FBH.
Never heard of Atlas Reed as a UK SAR provider. Maybe they run training elsewhere in the world out of a UK HQ.