I’m glad your going to tackle this project. Removing English ivy will be beneficial to your tree’s health.
So, first gather your supplies: Heavy duty pruning shears, a flat-bladed screw-driver, and then make sure you are wearing a long-sleeved shirt, sturdy pants and shoes, gloves, and eye protection. Working with English ivy can cause skin irritation for some people, and you may get dirt in your eyes as you remove it.
Plan to remove the ivy from just part of the tree, from waist or shoulder-height down to the ground. You will not pull the ivy down from the whole tree as doing so may dislodge insect or other animal nests, rip smaller tree branches, etc.
With your pruners (or a pruning saw, if the trunks are really large), sever ivy branches at waist-to-shoulder height. If you need to, pull them away from the trunk first by prying them off with the screw driver so that they are easier to cut. That will remove the ivy’s lifeline to the top of the tree; it will die off and gradually come down.
Then pull the remaining ivy trunks out of the ground. This is easier after a good rain or a thorough soaking. If the ivy has spread all over the ground too, you can spray the part of the ivy that has spread over the ground with Roundup or other glyphosate product, being sure to put plastic around the tree trunk to protect it from the spray. Ivy leaves are waxy, but young new leaves in early spring will take up the herbicide better. If all you have are older leaves, which have a fairly heavy waxy surface, apply the roundup after you’ve added a “Spreader-sticker” compound to it, being careful to follow all package directions.