Once completely dry the wood glue pulls off easily. It will leave no residue. If bits do break off and stick you just ball up some of the glue you have in your hand and dab it on top of any remaining bits. They will attract to each other. If you wish to be doubly sure you can always do a distilled water rinse and dry with a clean micro fibre cloth but in reality the glue will not have much adhesion to the base vinyl of the LP anyways.
I've done this a few times and have had good results, but it's too slow (drying) for me and as such not efficient enough. Today I use a Spin Clean cleaning setup that costs less that $80.00 typically and works well too.