The time for injuries to heal ranges from +1 to +4 (note that the + sign represents the red cross you can see in HT if a player is injured). Note that although the injury time is displayed in weeks, the actual healing time may vary. This is primarily dependent on the age of the player. The older he gets, the longer it will take to heal. The weeks displayed are just a standardized indication of the severeness of the injury. You can hire doctors to speed up the healing process. Physiotherapists will help players prevent injuries in the first place.
A player can not play as long as he's injured and his name will show up red in the match orders screen. If you do try to field him anyway, he will be replaced by a substitute as soon as the match begins.
Also injured players will get the training for the minutes they had played during the week. Once he's really almost recovered he gets a bandage next to his name instead of the red plus. Now he can play matches again and his name will show up black again in the match orders screen, but he will perform slightly below his capacities. Once the bandage is removed he's back to his old level, that is if he didn't get a major form crisis in the meantime.
Several sites and helper applications offer tables or calculators to predict how long an injury will take to heal. Here are some popular sites (only the first one is able to give useful results if you want to employ more than ten docs):
This site tells you how long you can expect a player to be injured based on the injury text in the match report.
Connection between TSI and displayed length of injury
There is a connection between the number of weeks Hattrick displays for an injury and the TSI a player has.
number of weeks=10*(1-tsi2/tsi1)
where tsi1: TSI when player is not injured (so usually at the time directly before the injury, let's say e.g. 9310)
tsi2: TSI now (e.g. 7000)
Hattrick would then display a +2. Injuries have a sublevel so it is actually between 2.47 to 2.49 as TSI has a granularity of 10. You should be aware that tsi1 is only known if there is no form change or training after the player was last healthy.