Ok so here is my take on this as a claims adjustor...
A few things aren't adding up and I would question straight away...you say you didn't see the cyclist u til impact, yet you say he was travelling at breakneck speed, this straightaway doesn't look good for you as how can you judge the speed of someone approaching you if you haven't seen them.
You were travelling at approx 15mph at the point of impact, this tells me that you didn't stop before entering the roundabout.
By your own admission you looked right once, and were then looking straight ahead therefore haven't check properly before entering the roundabout.
You have hit the rear wheel of the bike with your nearside, this shows that the bike must have been already established on the roundabout before you entered it.
You have admitted liability at the scene, granted that mean jack all as any decent liability adjustor will argue that you were in shock following the accident.
You already know the cyclist has a witness in their favour and you have no independent evidence of your own, therefore absolutely no way of fighting any liability disputes.
The highway code states that you should be aware of possible vulnerable road users and take extra care, this will go against you.
Also, speed is not an indicator of liability, particularly if that speed cannot be proven and as mentioned above you stating that he was travelling at breakneck speed suggests that you had seen him.
Ultimately, you will be liable for this and your insurer should have told you this right off the bat rather than give you false hope of a split liability.