Hi Tracey - I don't believe it's the end of your tree but just might weaken it in the interim - think you have done it a bit late and the sap is already rising:
You have nothing to worry about.
I have seen this often in my own garden. I seem to recall the the drip can continue for a very long period, possibly months. The only problem that I'm aware of is that this sticky stuff can drip onto you. It can be almost like rain.