In the highway code, the cyclist is instructed to dismount at most types of crossing and proceed on foot, however, this is not always the case, a cycle lane for example or a combined cycle/pedestrian path controlled by crossing lights would be an exception.
If you had struck the cyclist, then the cyclist might bear most of the responsibility, but do not take that for granted.
Drivers of all motor vehicles must give way to pedestrians, and cyclists, even if said persons are breaking the law.
There obviously limts to this, if some pedestrian/cyclist happens to move out in front 2 feet from your bumper without warning when you are driving along at 50mph, then you really didn’t have any options.
The police would investigate a pedestrian/cyclist collision with a motor vehicle and will decide wether you had reasonable time to stop, that the movement in front of you was reasonably predictable or not.
In your case, which amounted to a game of chicken, you had perfect sight of the cyclist, you could reasonably expect that person to move into the road (they didn’t in this case but it was reasonably forseeable that they might).
The police would probably do you for driving without due care and attention - it would be up to you to show in your defence that there was no other alternative, and for the police to show that there was something you could have done.There is every chance the cyclist would also be done for riding without due care and attention by the police too.
That’s for the criminal side fo the law, as for compensation, that’s rather differant, you had right of way, the cyclist did something stupid and any claim would be countered quite easily, you might even be able to make a claim from the cyclist if you got off with the ‘due care and attention’ charge.
It boils down to this, you do not have the right to collide your vehicle with any other person, or other vehicle merely because you have the right of way.