Copy and lasted for direct from he FA website...
The referee:
allows play to continue until the ball is out of play if a player is only slightly injured
stops play if a player is seriously injured and ensures that the player is removed from the field of play. An injured player may not be treated on the field of play and may only re-enter after play has restarted; if the ball is in play, re-entry must be from the touchline but if the ball is out of play, it may be from any boundary line. Exceptions to the requirement to leave the field of play are only when:
a goalkeeper is injured
a goalkeeper and an outfield player have collided and need attention
players from the same team have collided and need attention
a severe injury has occurred
a player is injured as the result of a physical offence for which the opponent is cautioned or sent off (e.g. reckless or serious foul challenge), if the assessment/treatment is completed quickly
ensures that any player bleeding leaves the field of play. The player may only re-enter on receiving a signal from the referee, who must be satisfied that the bleeding has stopped and there is no blood on the equipment
if the referee has authorised the doctors and /or stretcher bearers to enter the field of play, the player must leave on a stretcher or on foot. A player who does not comply, must be cautioned for unsporting behaviour
if the referee has decided to caution or send off a player who is injured and has to leave the field of play for treatment, the card must be shown before the player leaves
if play has not been stopped for another reason, or if an injury suffered by a player is not the result of an offence, play is restarted with a dropped ball
I think the section in boldness is what we are looking for. As long as the referee doesn't deem the treatment as taking too long and the offence that caused the injury is a caution or a sending off the player doesn't have to leave the pitch.