If there is a trained 1st aider in attendance then give them the opportunity to do their job. Even if you are a qualified doctor then you may find their training in 1st aid is better than your own.
If you are yourself a qualified 1st aider, and no one else around is then - if it were me (and it has been) I would take the role of providing 1st aid.
The main principle of 1st aid is that you must not make the situation worse. You must ensure that there is no additional danger around and that they have a pulse (best one to use is in the neck, under the jawbone and beside the adams apple -practice finding it on yourself and significant other) and are at the very least breathing and with clear airways. In rugby injuries you must always be aware of neck and spinal issues, if ANY doubt then DO NOT move the casualty; you can carefully cover them to keep them warm.
Luckily it is unusual to see a deep a heavily bleeding cut o a rugby pitch, even a laceration from a broken stud probably won't result in a really serious loss of blood. Pads, Pressure, and Elevation should suffice. If it is bleeding heavily and a pad gets blood soaked put another one on top, don't remove the first one.
Breaks are more common. Pain, swelling and loss of mobility may be the indicators (though not always). If its a limb try to ensure it's kept as immobile as possible. If it's ribs don't try to immobilise the upper body, and don't apply any pressure. Remove casualty from the pitch if "walking wounded".
In all cases make sure someone calls an ambulance. Make sure that person knows where they are, addess, ideally post code (for ambulance satnav) and how to get to the site. Don't let the casualty drink and certainly don't provide food - if they later need an anasthetic this may delay the time when that can be given. (Dr Stu may care to comment)
The only situation in which you may be forced into taking some definite activity beyond making them comfortable is if their airway is blocked or they have stopped breathing. If the airway is blocked, mud, false teeth, oppo hookers ear, whatever - try to get it out, be very careful not to push further in.
If pulse/breathing has stopped then the only thing to do is to give CPR.
Frankly I think this should be taught in schools and everybody should know how to do it, and get the opportunity to practice on a model (plastic, not super). If you nor anyone else can do that then they are going to be in bit of a hole. No pun intended.
I have heard the dire warning about being sued - frankly, so long as you interfere at the minimum level you can then I think, in the UK at least this is most unlikly to happen, and even less likely to succeed. I think our courts are far more sensible than, for example, US ones.
This post is not intended as authoritative guidance, nor is it in any way Hampshire Society policy - it is a purely personal view based on my own first aid experience and training in the Army, and later in civilian life.