When a double decker bus stops in front of you, what do you do? Is it safe to attempt to overtake the bus?
The Highway Code comes in handy with this question...
In London drivers of all buses are exceptionally well trained and likely to be highly experienced. They know every part of their route and have 'dealt with' plenty of cyclists before. They will always indicate when they are pulling over and indicate again when they are going to pull out. It is highly unlikely that you will be in their 'blind spot' at any stage of your overtaking manoeuvre.
It all depends on the traffic situation, how much speed you have and how important your journey is. Sometimes you can look through the back of the bus and see how many people are to dismount/catch the bus - if the stop looks like a busy one then you should overtake the bus, in part this is to clarify your intentions to road users behind you that may be 'waiting' for you to overtake before they do likewise. Conversely, if the bus is only dropping off one person, and, if the terrain does not suit you, then it can be advantageous to let the bus go, to 'draft' it and get a good, maintainable tow along the road.
Some bus stops, e.g. on a narrow road with a curve to the left may not allow you to see the oncoming traffic. It may be advisable to hold back in these situations, even if the traffic coming the other way is bumper to bumper. This is because a motorbike or other cyclist may be using the gap between the bus and the cars coming the other way and, once you have committed to overtaking the bus, you cannot duck in to the side again.
There is the Highway Code and you are supposed to let bus drivers pull out. Normal car drivers know this better than most cyclists although not all of them will let the bus out first as they do not want to be stuck behind it. As a cyclist you do not necessarily have the same need to overtake as the bus is likely to be faster than you between stops.
Hence, as a general rule, you should overtake the bus if the general situation is safe AND the bus does not have its right hand turn indicators on. If the bus does have the right hand turn indicators on and you overtake then that is against the spirit of the Highway Code.
Agree very much with Mathew's answer but would add a few extra points.
It is often the case that in city centre traffic - particularly where there are cycle lanes - that you will be much quicker than the bus between stops and traffic lights as you will not be limited by the queuing traffic as much. In the city centre there are a few other things to take into account. Has the bus stopped in front of you because of a bus stop, or because of queuing traffic? If you can't see what is in front of the bus, it may not be safe to overtake as you may not be able to pull back in to the correct lane.
If there is a cycle lane and there is queuing traffic in front of the bus, you may want to use the cycle lane to undertake the bus (assuming it is not at a stop). In this case, make sure that you can see a clear cycle lane in front of the bus and that there are no left turns coming up. Even then be extremely cautious. Undertaking buses (and lorries) which are about to turn left is the main way cyclists are killed in London. Untertaking you can easily end up in a blind spot and because of the turning circle of a bus (or lorry) you will be crushed if they do turn left.