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I don't know if any of these are open at the moment, but some seem to be able to do takeaway and/or deliver. It's hard to tell from most sites, though, as most don't have COVID-19 updates.
Ruby's still seems to be open for on-line orders:
https://rubysoflondon.com/product-category/cakes/
Their main outlet seems to be Greenwich Market, which I'd guess like all council-run markets must be closed at the moment.
The Hummingbird Bakery does vegan cakes, and although the latest blog post says it's still open for takeaways, elsewhere on the site it seems to suggest that it isn't--perhaps best to give them a bell:
https://hummingbirdbakery.com/search?q=vegan
The Canvas usually does cakes, but I'm not sure they're doing anything at the moment--they say you can order things off the menu, but cakes don't seem to be on the menu (the only one they have on-line is the December 2019 menu):
https://www.thecanvascafe.org/vegan-cafe
This says it's closed, but there's a phone number:
It's my girlfriend's 30th coming up and I want to get her a cake. Is there anywhere that does vegan cake off the shelf? I'm shit at baking so I guess the alternative is getting a local bakery to knock one up especially, there's a good one here in Leeds but I've already spanked way too much on her present so want to keep costs down!