Hi, OP. I sympathise. We had to cut down five massive trees when we moved into our new home, they were literally filling our (small) back garden and the house was so dark. We kept one silver birch, a 40 foot pear tree and started again with the rest of the garden. I felt dreadful about it as I love trees, but two of the silver birch were dead and leaning. One of our neighbours came round and yelled at me, as the trees had afforded them privacy, then they yelled at my mum when she came round. It was absolutely awful. The other neighbours were delighted as they hated the leaves and mess and their house and garden were dark too.
Anyway. I strongly recommend you actually nicely talk to the neighbour- removing the trees might have not been a tree-hating action. I have replaced a hideous enormous badly lopped conifer, with a series of lovely native trees. Yes, taking time to grow, but much better for local wildlife.
Do not plant leylandii, you will regret it as the cost and hassle of keeping them in check rapidly becomes prohibitive. And definitely don’t plant bamboo, it will run away and become a massive invasive problem - the neighbours will detest you and complain. Rightly so. Plus it is not cottagey at all.
I would buy hawthorn, rowan, birch, if you want trees. Interplant with some slower growers, like a flowering cherry, a crab apple and a holly. When the other trees get too big, the other trees can take over.
Portuguese Laurel could also be a quick win although it is quite boring, however it is easy to prune, grows fast, is evergreen and is more shrubby than tree so might be ideal. Doesn’t mind shade, either.
If you don’t fancy trees, then I’d plant ceanothus, lilac, buddleia, sambuca nigra (grows fast, doesn’t have to be chopped down every year just prune it and tie it in to keep a dense shape). I’d let all these grow into tree-sized shrubs. You might not get the height you want, so then I’d interplant several red robins - if the latter are about 6 feet tall when you buy them, it is an investment that will pay handsomely as you can let them grow tree-form and within 5 years they will be about 15 foot.
Mainly do not despair! This is an opportunity to do something lovely with the end of your garden.