You can start by severing the stems at the base. You can do that now and it won't affect birds nesting. I did that a few years back and it took at least three months before there was any sign of the leaves dying, because they got all their moisture from their aerial roots. I was able to pull the stems away from the wall and trees about six months later, without any trouble. You will need to check carefully to make sure you have identified all the main stems.
Meanwhile, watch out for any new shoots emerging at ground level, and zap them with glyphosate. Young leaves absorb the stuff a lot more easily than the more mature leaves.
Birds can have more than one brood, so you should leave the main clearance until August time. You can then treat the ivy more liberally with glyphosate. By then you will be able to spot which strands you have not already disconnected from their roots. You should add a few drops of detergent to the glyphosate to help it stick to the glossy leaves. The glyphosate will continue to work right through the autumn, as the ivy will still be active at this stage. Use a hand spray and take care not to get any spillage onto leafy plants.
