What are some English/British given names that can survive intact against (cypriot-) Greek vernacular?

Approach 1. You need a name that can straightforwardly inflect in Greek, or that looks like something that straightforwardly inflects. That means a male name ending in -os, -is, -as, or a female name ending in -a, -i, -o.

Not a lot of English names do, but you’d be surprised. My uncle Andreas (Andrew) is rendered by my aunt in Greek as Andros. I believe Andros is a Cypriot variant of Andreas already, but it’s also how such grammatical assimilation can happen.

If an English male name ends in -a, -i, -o, stick an s on and you’re done. As in fact occurs for English names bused by Greeks already. Jimmy > Dzimis.

If an English female name ends in -a, -i, -o, you’re already done. Jenny > Dzeni.

If you’re not in that category, you can get creative, as Andros shows.

Approach 2. A name that can be translated into a Greek equivalent straightforwardly, because they’re cognate. There’s no shortage of names that show up in both English and Greek, because they are either Greek or Latin in origin (Philip, Nick, George, Luke, Mark, Lucy: Filipos, Nikos, Yorgos, Loukas, Markos, Loukia), or because they are common Christian patrimony as originally Jewish names (John, Elizabeth, James, Mary: Yannis, Elisavet, Iakovos, Maria).

Approach 3. A sound-alike name, which I don’t think really counts. The Greek diaspora is full of Athanasios that have renamed themselves Arthur, and Dimitrios that have renamed themselves Jim, and Kostas that have renamed themselves Gus (that was always a US thing, and didn’t happen in Australia: they stuck to Constantine > Con there). You could flip that, and turn Arthur into Thanasis, and James into Dimitris.