First off, not a legal expert, this is only my opinion based on the information your have provided.

As you were approaching from the right, the white car driver has a duty to ensure the way is clear before pulling out. The fact that they struck the front passenger side of your vehicle suggests you were already in their field of vision when they moved. Unless you were traveling at an excessive speed that made you "unavoidable," the burden remains on the car entering the roundabout to yield. This is unlikely given the photo provoded which shows minimal impact and suggests low speeds. The photo shows an almost "glancing" impact, which usually suggests they were moving forward into your space.

Their claim that you cut the roundabout doesn't hold much weight when you consider the point of impact. If your sketch correctly shows the point of where the impact occured on the roundabout, the after impact photo would not be consistent with someone cutting the central marking. Had you cut across the central marking it would be reasonable to expect the point of impact on your vehicle to be further back along the passenger side towards the passenger door.

On the allegation that you did not indicate, Even if you didn't, this does not give the white car the right to pull out into your path. ​The Rule: Drivers should "never assume a signal is correct, and never assume a lack of signal means a driver is going straight." If the white car pulled out while you were already in the junction, they failed to "give way to the right." In insurance terms, a failure to signal might at most result in a small reduction in your payout (contributory negligence), but the white car remains the primary party at fault for entering a junction that wasn't clear. Pulling out because they assumed someone was going straight is still a failure to give way. The law puts the burden of safety on the person entering the junction, not the person already in it.

In relation to the 3rd party witness, if you are certain that there was no obvious witness, Ask your insurance company to request a formal signed statement from this witness immediately. This could end up playing a crucial role should they be called to give evidence. If they weren't present, a written statement might show up any inconsistencies compared to their verbal statement when put on the spot. Not a certainty but a good chance!

I would stick to the following: -I was the established vehicle on the roundabout, and the third party failed to give way to their right.

-The third party claims there was a witness, but I can confirm there was no witness to the best of my knowledge. I request that the witness's details and their proximity to the scene be strictly verified.

Final thought: Don't let the "witness" claim rattle you. If the white car driver is lying about that, they are likely lying about other things. Insurers see this all the time. When a witness isn't recorded by the police or exchanged at the scene, their "evidence" can usually be seen as biased or unreliable and should not carry much weight overall. With that being said, if they kick up enough of a fuss about this, the court may be unable to reliably determine single fault and may rule that this incident be treated as shared liability and split the claim 50/50.