What "deadline"? There is no deadline, you just need to complete all the main scenario content up to 2.55 before you'll be able to move on to Heavensward content. None of that is going anywhere, it's not like the patch is going to drop and they'll go "ha ha, **** you, now you can't complete the storyline".

It's unclear whether the level cap will automatically be raised from 50 to 60, but even if it is, the existing main scenario quests probably aren't set up to give any XP at 50 anyway, and the amount of XP you'll gain from mob kills will probably be absolutely trivial compared to what it takes to level up without doing the new quests. Otherwise you'll just continue exactly as you are now until you finish the 2.55 storyline and can go to the new zone.