Do Greeks who came from Turkey in 1960 have a different accent?

1960 in the question certainly alludes to Istanbul Greeks.

There has been minimal attention paid to the dialect of Constantinople/Istanbul, because it was an urban dialect, and historical linguists were interested in the countryside, as more archaic material: Constantinople itself had all unstressed vowels, like Southern Greece, and unlike the villagers of Thrace, who reduced unstressed vowels—as Northern Greece does.

I’ve just discovered that Valentina Fedchenko of St Petersburg State University has written a paper on the language of Constantinopolitan Greeks in Athens, in 2007: Les Grecs de Constantinople à Athènes: perception d’une langue étrangère. (For any Greek linguists reading: yes, she’s one of Maxim Kisilier’s students. Looks like Fedchenko is now working on Yiddish.)

The shibboleths of Constantinopolitan, which I already knew about, and which (as far as I know) it shares with Thracian dialect are:

• Use of που rather than πως as a complementiser.
• Use of διω rather than δω as the subjunctive aorist for ‘see’.
• Use of accusative rather than genitive indirect objects (common with Macedonian and Thessalian Greek)

I’ll summarising what I’m seeing in Fedchenko’s paper:

• Lots of codeswitching into Turkish, even to the extent of putting Turkish inflections on Greek words
• Lots of French words—and dismay that Athenians are too unsophisticated to use those French words. (They should have been around a century ago.) Examples, to freak Greek readers out: mentalité, civilisé, dîner, cure-dent, quartier, vendeuse, politesse, garçon, demande.
• Resistance to assimilating linguistically, with some use of dialect to avoid being understood, but also much pride in their variant as more correct than the Athenian standard.
• Particular relish for dialectal archaisms (such as απίδι rather than αχλάδι for ‘pear’), which to them elevate the status of their variant.
• Continuing cultivation of katharevousa.
• There have been some recent dictionaries of the variant; I’m annoyed that I hadn’t heard of them.