Who has right of way at a staggered crossroad
Yes, this is a tricky situation. I think this section of the highway code should cover it, but is very light on details like this.

I think there are two ways to interpret the situation:

1. As two T-junctions a small distance apart, with you turning left and then right.

In this case it's a fairly clear cut - see rule 180. You've turned left, already on the main road and have priority over vehicles trying to join your lane, including ones turning right out of the road you're about to take.

2. As a normal crossroads, with you going straight on, the other car turning right.

For case 1, from Internet discussions like this from Yahoo and this from the RAC there seems to be little consensus on this. Many say that the traffic going straight on has priority, but then add that you have to give way to traffic coming from the right (as you would on a roundabout), which seems contradictory.

My opinion

I think this needs official clarification, but in practice it would seem sensible to interpret it like rule 170: "watch out for pedestrians crossing a road into which you are turning. If they have started to cross they have priority, so give way".

When at a give way line, all vehicles must give way to others not crossing the give way line. So if you arrived at the junction and there was no other traffic coming, of course you would start to cross. Being a bike you're usually slower than a car, so a car which later arrives opposite you at their give-way line should give way to all traffic already on the junction or the other road. 

Furthermore, from the look of it this is a box-junction, so the car must not proceed until its exit is clear. In the image below the green car attempting to turn right is in a very similar situation to the one you mention.



If you both arrived at the give-way line at the same moment, then given the above image, you might still have the priority (de jure), but it's not obvious and my instinct is to let the faster vehicle go first.

As others have mentioned, us cyclists have to be careful as collisions are usually more serious for us. Whenever cars do show you consideration, always give them a wave or nod of thanks, even if you have priority.