How much time, if any, should teenage swimmers spend warming up before a training session?

I think this answer comes in two parts.  Most coaches structure their training sessions to progressively prepare their swimmers for their "main sets,"i.e. the most intense sets that are the critical objectives for that practice.  Roughly, that gradual build-up would be 20-25% of that session's total volume.

Before that, when the athletes are standing on deck, dry and warm, their coach will ask them to "warm-up."  What is occuring there is moving the large amount of blood that is in our skin, through the internal organs, to the working muscles.  That takes about 8-10 minutes.  If the "teenager" is 18 or 19, probably 8-12 minutes would be appropriate.  If the teenager is 13-17, their attention spans and ability to warm-up independently are much shorter, so the coach will probably prescribe a 4-6 minute swim to "get the blood moving," then move into their series of preparatory swims leading to their main sets for the session.

I would agree with George. Generally if the session is early in the morning, the warmup should be more dynamic and active. Static stretches are less effective. Get them moving and "working" the different muscles groups to get that blood flowing well.