Honestly, I was disappointed with the Panasonic Lumix G 20mm f/1.7 II ASPH. Its sharp, but the autofocus is sluggish and hunts in low light, which sucks for London. Youre better off grabbing the Olympus M.Zuiko Digital 17mm f/1.8. It hits that street sweet spot and the focus is way snappier. If you want reach, the Panasonic Lumix G 42.5mm f/1.7 ASPH POWER OIS is incredibly sharp, though it might be tight for alleyways.

I've been using the GX85 for a few years now and honestly, you can get some killer shots without spending Leica money. The Panasonic Lumix G 25mm f/1.7 ASPH is definitely the best bang for your buck. I am super satisfied with how fast it locks focus, which is gonna be huge for you in London when things are moving fast. It is light, cheap, and gets the job done well without any fuss. If you want something a bit more premium but still under your $400 limit, I would seriously look at the Sigma 30mm f/1.4 DC DN Contemporary Micro Four Thirds. It is a bit bigger than the Lumix primes but the sharpness is on another level compared to the cheap 25mm. Plus that f/1.4 aperture is a lifesaver for night street photography or inside moody pubs. If you really want that wider 15mm look but cannot afford the Leica glass, check out the Panasonic Lumix G 14mm f/2.5 II ASPH. It is tiny, really sharp, and focuses instantly. It is one of my favorites for travel because it weighs basically nothing and makes the camera feel like a point-and-shoot. London can be cramped, so having something wider like that helps a lot with architecture. I have had no complaints with any of these on my GX85, they just work well and keep the kit portable. Just stay within your budget and you will have plenty left over for travel expenses.