I have just cut some branches from our weeping willow tree because they were overhanging a pavement The cut branches are 'weeping' water...Will this damage the tree

It will stop eventually and heal itself, it's just sap leaking out of the wound.

Basically, for future referance. You have 2 types of tree, wet and dry. Here are a several examples of both. Wet should be cut in the dormant season. Willow, Alder ( you'll find both by streams etc. ) Conifer and Birch. Dry woods can be cut any time apart from spring. They would be Oak, ash and beech. Cutting limbs away from a wet tree, ie. Birch in the height of summer could mean that the tree literally bleeds to death. This is ironic as people generally have any sort of heavy tree work done on Conifers in the summer. So there you go. I'm sure the Willow will be fine.