I'm only a retired controller, not a pilot, but I'll try to answer one or two of your points from personal experience.
Before VOR/DME became the 'standard' nav system, several other systems were tried, one being DECCA/Doppler. This was fitted to most of the BEA (British European Airways) jet fleet - I remember seeing it on Trident Mk 1s. I also remember when I was an assistant that in order to assist separating opposite direction aircraft, the pilot would be requested to 'fly DECCA left'. The DECCA/Doppler navigation system was put forward as the 'standard' worldwide but the American VOR system, shortly to be augmented by DME, was chosen instead. DECCA stations in those days (consisting of a master station and 3 slave stations), were already positioned around the world to provide navigation for shipping.
As to your questions:
The Tech log should have been read by the crew before accepting the aircraft so they SHOULD have known the VOR had been replaced.
Some companies required their crews to operate on one VOR and keep the other for backup whilst some required the in-use VOR on one and the next expected VOR to be dialled into the other.
I am unable to answer the third question.
The only backup to radar in those days was a pilot's report of position to a non-radar controller.
I've probably got some of the finer points wrong but it gives you some idea; maybe a BEA pilot from those days will see this and correct me.