Further to my previous post, I just add on a couple of other ideas that come to mind.
1. You might take pictures as it is now, for both future reference, and also it might be worth your talking to a local gardening club, who might have good information as to the best tree people around and also ideas of what might be grown in shade.
2. If you dont think it would stir up a hornets nest, you could speak to your council and check out the situation as far as they are concerned. I know there have been changes about very high hedges but it doesnt seem to apply in your case. However it might be useful to check how the local bylaws might affect it now or in the future. Then knowing you are not doing anything against the law, you can relax and enjoy your garden.
3. If your neighbour doesnt like it ,then you can either just ignore her where possible, or tell her what the council etc have said and offer her ideas for plants for the shade. Then go back to enjoying your own garden and let her get someone else to look after her hens. Oh by the way if she has a cockerel you can also mention that that is quite disturbing for you too, so that problems are not one way only, but that you try to live in harmony as you obviously have done until she came along. (Oh it might be quite useful for a little while to have ear phones in when in the garden. She doesnt need to know whether you actually have any music on or not!! )