It's short for "taking the piss out of (someone or something)." To make fun of, but in a light-hearted way that implies the speaker has the advantage of the target. In the instance above, it means "using an exaggerated British accent to lampoon people who genuinely speak that way."

The "piss" in question is seriousness, authority or importance (compare "piss and vinegar"), but you can take it from someone with the aid of humour. Piss-takes can be a form of speaking truth to power.