Happy to report that this is now solved.

The vibration was from the exhaust.
The rear silencer was vibrating which transmitted throughout the car.

The rubber hangers are worn( back left especially bad), allowing a tiny bit of up/down movement in the back box. I have temporarily fixed it by wedging something into a void in the rubber but have ordered all new rubber mounts.

The pulsing feeling when reversing slowly was the runout in my front brake discs causing uneven brake effort.

It’s nice to have a smooth idling car again and a very cheap fix.