I've noticed Tony Blair's use of this discourse feature, yes, but I don't have an impression that other people are using it any more frequently than it used to be. I haven't studied the matter, but - as always, when questions of this kind come up - a visit to the online OED generally provides some illumination.
There we find (under look v.4, 'idiomatic uses of the imperative') that this feature is as old as Old English. It turns up in Aelfric, and throughout history we see it in a variety of forms, such as look here, look you, look'ee, and looky here, as well as simple look. Shakespeare has 'Look you how he writes' (Henry IV Part 2) and 'Look thee here...' (A Winter's Tale). The tone varies greatly, from affability to annoyance.
The OED definition of look here is interesting - ' a brusque mode of address prefacing an order, expostulation, reprimand, etc' - as this very much relates to the Blair usage. My impression is that he uses look only when he's irritated by the way an interview is going, and wants to restate or amplify a point. He isn't saying to the interlocutor 'we're together on this' but 'you've got me wrong' or 'you're pushing me in the wrong direction'. Take these examples from a Newsnight interview with Jeremy Paxman in 2013 (transcript available online): the tone is moving towards exasperation - explicitly so, in the first instance:
'Oh rubbish. Come on Jeremy, look what do I stand for?'
'I don't know. Look, I am a Christian, I believe in it, but...'
'BLAIR: ... it doesn't inform every political decision I make in a very narrow way. PAXMAN: It doesn't? BLAIR: Look, I'm a person, an individual...'
With a less aggressive interviewer, the looks virtually disappear. Well and you know are his main discourse features. In a 20-minute interview with Sian Williams on the Andrew Marr Show (9 February 2013, also viewable online), he hardly uses look at all. We don't hear one until 9 minutes in, and then it's almost inaudible in the middle of a sentence ('look, it's very hard...'). There are two more medially-placed examples, neither especially prominent, and we don't hear the prominent sentence-initial look until almost the very end of the interview (18 minutes in, when his voice is clearly tiring and the interviewer is still pressing a point). We then get two instances in quick succession. I think he wants the interview to end - and soon after, it does.