Slang? Possibly, but you can go anywhere in the UK and ask "What's on the telly tonight?" and people know what you're asking. I'd claim that it's informal rather than regional slang. There's even an L.A. company using it as their name: https://en.wikipedia.org/wiki/Telly_(company)

(I was addressing the parent's claim of 'But English also has no word for "television"')

Yep you provide a great example of a word used as regional slang for the word television.

The word telly is not in common usage in the United States. It’s understood here to be UK slang for TV.

Your example is largely irrelevant; I wouldn’t call a spyware TV founded by Russian born dude a cultural touchstone.

Regardless of origin the word television is an English word now. The ability to adopt loan words from other languages is one of the many reasons English usage is so widespread.