There are 'hit and run' accidents where the driver notices that nobody has noticed and legs it, to leave the victim all alone, however, these are the exception. By and large most accidents do gather a small crowd in a short period of time and one or two individuals will take the lead in helping as best as they can. Therefore, by the time you arrive at the scene or shortly thereafter they may well be people better able than yourself to look after the injured cyclist.
The 'how's my bike?' question will tend to be trivialised by those 'phoning the ambulance' and those that arrive in the ambulance, however, to the one in pain this is the top question even if it seems a bit silly given the state that they are in. Therefore, as a cyclist, the best thing that you can do is to be completely honest to them about the state of their bike, and in detail. Even if it is bad news it will put their mind at rest.
If you have a camera on your phone then take some pictures of the scene and the bicycle. With that done you can take responsibility for what happens to the bike next.
If it looks like the ambulance is going to be used that day (e.g. for a check for concussion) then ask them if they have a relative able to come and collect the bike. Offer to wait with the bike or ask a local address if they can look after it until it can be collected. Obtain and return D-lock keys to secure the bike if required. Locked up inside a building but not to anything means that their relative will still be able to collect the bike, the cyclist will also know it is not going anywhere until such time as it is collected.
If there is no ambulance trip needed then let them know that you can help them with a second opinion on whether the bike is okay. If you have tools to straighten the bars etc. then offer your help with that.
Be aware that the bicycle might be at fault rather than the driver, the rain, unicorns in the street or anything else. Check the bike for something lethal such as a broken front mudguard that may have got caught in the fork and thrown them over the bars. If you do establish the responsible cause then the cyclist will want to know about it to avoid a repeat incident. Tell them your suspicions diplomatically.
Clearly there will be times when you will be first on the scene and that you will have to do more than look after the bike. As a cyclist you have the benefit of being able to park up instantly and you are also better suited than pedestrians to take control of the traffic situation. Unless you are in the middle of nowhere an ad-hoc team will materialise rapidly, play to your strengths as a cyclist in this team and be the eyes and ears for whomever takes the lead in looking after whomever is hurt. Remember that you are very mobile on the bike and you can go-get stuff in the immediate locality quicker than those on foot or in a car.
I have been fortunate in never having to get involved in actual first aid due to someone more proficient than myself being on the scene in a short space of time. Hopefully another contributor can post the ABC's of first aid as a complimentary answer to this question.