Looking at the traffic orders in The Gazette, there's an exemption for "maintenance, refuse and other essential vehicles", so it should probably be:
1) vehicle=private + bus=designated + bicycle=designated
2) access=private + bus=designated + bicycle=designated

I suspect the "no through route" signs are intended to stop council vehicles using the road as a short cut, rather than being directed at the general public.

https://www.thegazette.co.uk/notice/2769434