I think I remember the Blair usage, and it was notable because it was probably one of the very few usages of the phrase in the original meaning. But it's commonly used in the modern meaning even in UK newspapers.
A piece on it with some numbers on usage
http://www.guardian.co.uk/media/mind-your-language/2010/may/...
"After more than 35 years in journalism I can recall precisely one occasion when "begs the question" was used to describe a logical fallacy, by the philosopher and sometime Guardian columnist Julian Baggini."
The Telegraph has something like 147 usages in the modern meaning and none I could find in the old, I found 1 usage in the old meaning in about 70 in the new meaning in the Financial Times, there's a couple usages in The Times in the modern sense and none in the old.
Here's the singular usage of it in the old sense across these three publications
http://www.ft.com/intl/cms/s/0/e6f9f436-b788-11df-8ef6-00144... (interestingly I believe Caldwell is an American)
If there's more usages in the older sense then I just missed them it's so rare.
I've never actually heard it said out loud in the old meaning and the only time I'm even aware that there was a prior usage of the phrase is when some pretentious, pedantic, proscriptivist with a dangerously improper understanding of how languages work gets involved and tries to correct how other people use the phrase with a hopeless archaic usage that's long since fallen out of fashion.
I don't think I've ever heard the phrase "raises the question" in my entire life. I'd say that phrase, now that I roll it around on my tongue a bit, sounds hopeless awkward and nonsensical. You can't actually "raise" a question, you can raise awareness about a question. You can "ask" a question, or you can have a situation that requires a question be asked -- e.g. the situation can "beg" the question be asked. But raising a question doesn't really make any sort of sense in any possible usage. At any rate, if "raise" is meant to be used in the same way that one can "raise" awareness, then the meaning is still different than begging a question be asked. For example,
"I would like to beg for your awareness about the plight of the African tree frog."
has a more urgent meaning than
"I would like to raise your awareness about the plight of the African tree frog."
I don't just want you to have some slight exposure to the frog, it is of great importance to me that you learn about this frog and his plight. It's so important that I'll beg you to learn about it. Something that is reserved only for people in a desperate situation.
If these meanings of the two words are considered, than a situation can "beg" or "raise" questions with the same sort of urgency qualities as a person.