… A very good question, Clarissa!
On the one hand, not much, because English is in every household, though the telly and the interwebs. Now, where to find evidence for this?
Journalistic Greek is awash with ill-fitting calques from English, and syntactic loans and semantic field readjustments too, because the journalists spend their time reading (and paraphrasing) what they read in English.
… Except, 150 years ago, they were doing exactly the same with French. So Journalistic Greek is not more globalised now than any elite European lect was a century ago.
There’s a lot more evidence of this on the street. Calques like “give you a call”, say. Effortless codeswitching, even if it is heavily accented. This stuff is definitely going on now, and it didn’t even need to wait for the internet; the globalised mass media was enough, and that’s been in place for 50 years.
But still, that’s comparable to the French superstrate on Russian, or any number of other superstrates. A sprachbund, I think, needs more: it needs daily bilingualism in everyday life and the hearth. It needs people actively speaking English alongside Greek at the shops and at home, not just passively consuming it: that makes it much easier for the linguistic structures to commingle and be remoulded, outside of the occasional calque, and for the commingling to reach into morphology and in-depth syntax.
I don’t think we’re there yet. I think it’s not that far off though.