"I think on the whole I'm really considerate tbh. FWIW any underwear, risque or otherwise, is pegged out of sight of her garden so I'm sure I haven't offended her in that respect.
I can't help the fact that I have lots of washing to dry and the best way to dry it is outside. There is none out this morning but of course it's raining, when it clears there are two loads ready to peg out but that's how it is. I'm sure that she must have had to peg out more than once or twice a week when she had her four dc at home but I suppose you soon forget what life is like with a big family."
OP you do sound very nice and we can all tell you are considerate.
I do think you could help having so much laundry though. To be clear: I am not saying you should change a single thing. However: we do have a degree of control over how much laundry we do. Pyjamas worn more than one night in a row, that sort of thing.
Basically, I think she is struggling to understand the sheer volume of washing there is because in her day, she and her family probably owned a lot less clothes, might have shared beds, changed bedding less frequently, et cetera.
You've also said that you don't use the garden for laundry when the weather changes. So I assume you have an alternative, indoor drying system.
Again: I am not saying you should change a thing about the way you do things. BUT: from your neighbour's perspective, she has compromised up til now by not saying anything for at least a few years [I assume from your mention of xmas cards in your OP]. Rightly or wrongly, it's clearly taken a lot for her to summon the courage/energy/whatever to make this request of you. And despite what people on here are saying, she is not suggesting that you never use your garden for laundry. She has asked you to meet her less than halfway, by leaving one or two days without laundry.
I'm just saying, it would be worthwhile trying to see it from her point of view. It doesn't sound like she has been hostile or rude to me. She is hoping that, if it's not of too much inconvenience to you, maybe things could change a little, because perhaps it hadn't occurred to you that it was impacting on her. I reckon anyway!