"The power went out" is the classic American English phrase, at least since electricity became standard in most US homes (which was not everywhere until the 1950s, although Hawaii had electricty before Washington DC did). We also say: The electricity went out. The electric lines are down. (If they are the old fashioned electric lines on electric poles.) I do not think we would say "cut off" unless the city or power company deliberately cut off the electricity due to the dangerous situation. As for the internet: My internet went out: This is by analogy with electricity. My interent service isn't working. My internet service is broke. (This is colloquial.) I can't get online now, because my (internet) connection is down. I'm sure there are a dozen more ways to say the internet is down. Probably because it has not been around long enough to get fixed phrases used by everybody.

I agree with Alan. In the US, the most common way of saying that is "the power went off" or if you call home and someone wants to warn you about the lack of electricity, I use "we're out of power" For internet I say "my internet is down" or "internet is working". As a joke, sometimes we use "the internet is broken". It all depends on who you are talking to, formal or informal.

Can't we say: I've been powerless?