I have no idea why this weird pair of wood pigeon elected to nest where they have. It is probably the very worst position. There are also crows and maples in the garden, which often perch on the roof, overlooking the nest site (See photo below). Given half a chance, they would steal a fresh pigeon egg or two. The nest is at head height, above the dividing wall between us and our neighbours, so humans are regularly moving round and about the nest. And of course squirrels have been known to visit, searching for food, such as a fresh egg.
There is plenty of information on the internet in reference to wood pigeons. The following link is for wikipedia, which is detailed enough.
https://en.wikipedia.org/wiki/Common_wood_pigeon
The nest was completed about 3 weeks ago. That was painful to watch. It seems that the nest construction is a joint effort between male and female. It was a case of the blind leading the blind. One or the other would arrive with a 'suitable' twig, firmly held in the beak. The next 10 minutes were spent, trying to weave the twig. In and out of the nest, turning and twisting the twig to different angles, until finally it fitted. They got there in the end. Not a pretty sight, but its home. The female must be happy with the build quality, and that's all that really matters.
I have no idea when the egg(s) were actually laid, but I would guess that it is between 2 to 3 weeks ago, from today. Apparently the eggs hatch after 17 to 19 days, which means that hatching is imminent. Here's hoping.
I am in the process of setting up a CCTV to allow video streaming in real time. I will advise soonest and hopefully be able to provide a link, for you to watch at you leisure.