I studied a BEng EE course here in the UK, and got a job shortly after graduating a couple of years ago as a junior dev.
As others have said, the important thing is really just having something to show - I had this from taking on a little extra work while I was studying for some extra cash.
I think learning C and later C++ was really useful, though this may be a personal thing, I guess it was my first exposure to real programming. I often catch myself thinking how things would be done in C++.
Personally I dont feel that not having a CS degree is a drawback, I think there are some benefits to coming to the field from a different angle. But take this with a pinch of salt, I'm still early in my career, and web development is a broad term.