If you want to fit in with the locals, The North Face is very popular in the UK. I don't think I've seen any of our friends from North America mention it on here. I don't have any particular recommendations for jackets for ladies though.
Patagonia is going to score some points over Eddie Bauer around town too. It's another quite fashionable brand over here. Eddie Bauer isn't really sold over here I don't think.
I have an older down jacket by a brand called Rab, which is quite a popular hiking / mountain brand over here. It's worth Googling their range of jackets for the rain too.
Personally, I think trench coats and totally waterproof / Goretex etc is way overkill for London in August. The weather is unlikely to be that bad. Sustained rain and wind would be quite unusual in my experience. By August in 2022 everything was parched and dry in London. We had a few rainy days last summer, but nothing I'd call significant. When you're in town it's not difficult to keep dry if you need to.
I wear a Blocktech Parka from Uniqlo almost every day. It's waterproof enough and a bit more stylish for town than the hiking brands. It has that technical look, but I like that the sleeves are cut in a way to hang in much the same way as a suit jacket in length and width.