Jewelry store recommendations in London

Hi all, my partner and I are going to look at engagement rings soon. We live in London, but we didn’t grow up in the UK so honestly we don’t know that many jewelry stores in the city. We are planning to hit up the classics like Tiffany’s and shops in Harrods, and have done some research for smaller reputable shops, which I would prefer, but it’s a bit overwhelming which ones we should go to and it seems in the UK you have to make appointments instead of just popping into the shops to try stuff on. So if anyone has any recommendations that would be helpful. We are looking for a natural mined diamond.