If the trees need trimimg by 10ft they are probably a bit overwhelming for the neighbours. As you can see how it is affecting their light, and they have had to write you a letter, I doubt there is much to be gained from putting this off any longer. I suggest you pop round and have a chat with them.
Unfortunately these trees are your responsibility and they do not have to be sympathetic toward your financial situation, but your pregnancy will be quite obvious to them if they see you in person. Be bright, smiley and as friendly as you can and see if you can negotiate a contribution. Sadly if they don't bite, you will have to pay. Get a few quotes and expect to pay about £250 for the job.
Whatever you do, don't let it go any further as it will be stressful; you will fall out with the neighbour and the council will be after you for costs.