I'd definitely second the advice to check whether or not the world class institution actually delivers world class supervision and facilities. Sometimes the biggest names (universities & supervisors) are actually not the best for a PhD.
And also depends what you mean by second rate - I'd be hesitant about suggesting you did a PhD in a dept that didn't submit to the RAE, but if by second rate you mean Russell Group / 1994 group as opposed to Oxbridge then the difference is unlikely to effect your future career.

Thanks for your well-considered replies.

My first instinct is to delve a little deeper into the reputation of the potential supervisors.

Having said that, though, it's not my intention to follow a long-term academic career: the purpose of my gaining a PhD is to fast-track access to the higher professional roles in my specialist area. And here's the rub: if I'm not aiming to live in academia, perhaps the association with a high status collaborative partner (albeit in conjunction with a weak institution) will be more significant to my future career than being supervised by an academic at a top-notch university.