Give way to car on opposite side of roundabout just in case?

[Chromegrill]

The Highway Code says give way to cars already on a roundabout as you never know if the clown to your right is signalling correctly.

What about if you enter a roundabout at precisely the same time as a car approaching the roundabout coming towards you in the opposite direction however, and not signalling a right turn? Should one (or even both of us) stop and wait for the other to move, or do you go over and hope they aren't about to swing right into you?

Happened to me today. I slowed down to approach a roundabout - good visibility, crossroads with fairly small grass covered roundabout in the middle. Car approaching from the opposite direction entered the roundabout at exactly the same time as I crossed the threshold, neither signalling nor slowing down as it did so. I'm already half on the roundabout when to my horror I see him coming right at my offside at a pretty hefty speed as he suddenly puts in a sharp right hand turn. No signal obviously. I braked an emergency stop to let him whizz past and thankfully the car behind didn't crash into the back of me in the process. But it was a close shave.

Question: is there anything I should have done differently given I had no indication that he would come rushing towards my offside at the point I was starting to come onto the roundabout?

Supplementary question: does your answer above change given the car in question was an Audi A3?

[Chromegrill]

Thanks that was my logic - but it failed on this occasion. Had I not slowed to approach the roundabout I'd have crossed it before the Audi reached it. In slowing down it delayed my arrival to the same time as his. As he was belting it along he was already about to turn as I was passing the white line. He didn't indicate and turned from the left hand lane so until he turned I had no way to anticipate such a manoeuvre. I guess I'll put it down to experience - and be extra vigilant for hoodlums in Audi A3s.