2 years ago I moved from the games industry into the finance industry. My role now is varied and would probably come under either quant-dev, e-trading or algo-trading, depending on how you look at it. I was hired with zero financial knowledge, on the basis of my maths and computer science skills, and it has so far gone very well.
I was lucky in that I had contacts in the industry; without that it is probably harder, but it is definitely doable. If you've got strong maths + computing skills then there are a lot of possible roles for you.
Web skills aren't terribly useful (although not worthless); a lot of things are different in the financial world where for example correctness, latency, etc. are much more important. Financial software tends to be written in either low-level languages (e.g. C++ for execution stuff), or high-level functional languages (Scala, Haskell, OCaml - see e.g. GSA, Standard Chartered, or Jane Street). Some places use Python for analysis stuff, though R/Matlab are very common too.
If you want to into an actual finance role (Trader, or to a certain extent Quant (as opposed to Quant-dev)), that's possible but you'd probably have to go via a technology role. That's your way in.
I've seen a tiny bit of Scala, but not mainstream adoption. Python is pretty popular in a few banks especially (EG BofA).
A lot of banking jobs ask for "Banking Experience", but not all. Also, I would think doing a few finance/quant courses on Coursera wouldn't hurt.
It's not mainstream, but it's also not niche either.
Betterment, Wealthfront, SigFig, even RobinHood... all are companies that bring tech into the financial industry.