I know both very well. Grew up in Athens and lived in London for 14 years. Will try and answer all your Qs but let me know if you have any other concerns/questions.
Language - the vast majority of Greeks speak some English. Enough to get by. They all do it at school. How good they are depends on how good their teacher was. As with most things. Still, my English DH speaks about 5 words of Greek and gets by fine when I'm not around. Any attempt to speak Greek is met favourably and you'll pick it up pretty quickly once you're there. Road signs and many other signs are in Greek and Latin characters. People will help you out if you are lost or need something.
Healthcare - as long as the UK remains in Europe you are covered by that. Otherwise who knows. There are some costs associated with some healthcare and I can get you more info on that if it is a concern for you. Greek healthcare is OK. My mum has no private healthcare and does fine with state care. My dad has private Greek health insurance at the cost of about 1000 Euro a year and is currently undergoing chemo for an incredibly rare cancer at over 4000 Euro a round every 21 days all 100% covered by his insurance. So that is an option.
Life approach - I almost called this frustrations. It is a mind set. You need to accept that things are slower and don't work or happen as you might like. But also that this is not a measure of how much people care about getting a job done for you. Greeks work hard - regardless of what the UK tabloids would have you believe - they're just not intense or competative people and they are limited by the slow moving machinations of what is a bit of a decrepit system. Parts of life are 21st century but a lot is still lingering in the 80s. I find a weight lifts when I get off the plane there. I miss the slow moving, chilled out way of life. It suits me more than intense, high octane living. Where I live now is more bonkers than London so I am feeling the pace.
Life - you can't beat it as long as you have a bit of money ( enough to live). It is fun, relaxed, warm, food is amazing. Heaps of art, culture and history. Greeks are emotional people and that has ups and downs but they are warm and friendly and you can feel it. Yes, Athens is a busy place and people aren't fluffy but if you have an issue they'll step in to help. I always feel safe. Someone touched my bum on a packed train once and at the next stop when the doors opened the entire carriage (led by your classic Greek granny in black) shoved this man out of the train. People talk to strangers. If you go in a local shop expect a chat - even in downtown Athens. There are strikes and you won't always hear about it until you try and get a taxi in the morning and find their union is taking the morning off. That can be hard but people work around it with car pools, public transportation etc. That said, the sun shines nearly every day. You need sunnies in the winter even on freezing days. Travel is amazing. It is a small country but the landscape is so varied and it is beautiful. Just thinking about parts of it makes me cry with the beauty of it. It is so much more than islands and beaches. It is so mountainous and in places it is still so wild.
Cost of living - You can do your shopping cheaply by buying fruit and veg from your weekly street market, etc. Bread is cheap. Meat can be pricey. Alcohol is reasonable. Petrol is pricey. Public transport is cheap and good. Eating out can benvery cheap or not at all - depends where you go and what you order. There's a lady who does home cooked Greek food near my parents' house who crages 5 Euro for enough amazing home cooked food to feed two for dinner. But you can also get fancy food at 100+ Euro. Rent is reasonably cheap due to the crisis. A decent 2 bed apartment in downtown Athens is about 500-1000 Euro a month depending on what you want. About 400 in the suburbs.
Challenges - the economy is not great but is improving. You now have a country trying to rebuild as people have accepted the government are a bunch of morons and we will just have to go it alone. High streets in many areas of Athens are looking healthier than they have been for a while. There is a lot of creativity and young people are trying to set up businesses etc as there are no jobs for them. There is some investments coming in now. I will say though that paperwork is still a nightmare - partly because of name spellings etc. Partly because beaurocracy reigns strong and red tape is their favourite. That said, we are getting there. Tax papers are going online. Land is finally being mapped etc so things are getting better there. It is slow and frustrating.
EU - Greece has said they will allow UK citizens to stay as there are so many Brits in Greece and more Greeks in England. As long as England doesn't kick them out Greece won't kick the English out. You would probably need some kind of visa etc but I would imagine that will take some time to sort out.
My mum is English and has lived there since the 70s and although it frustrates her she adores it still. She and my dad ran a successful business together for years and have had a wonderful lifestyle. The crash was hard but it is getting better for them now.
Greece is European but you can tell it is only part of the way there. And Greeks like it like that. It isn't sterile or efficient. Pavements are a mess, potholes galore. The driving is terrifying etc. But Greeks like all that or don't really pay much heed to it. The rough and smooth is part of our culture. We know it's a bit pants at times but we muddle through. Life is highly politicised. Money and politics is what we all talk about around the kitchen table. It is very un-English in that way. There is racism but not in everyone. It varies. If you are English you won't notice it. But it is the one aspect of Greeks and Greece that makes me angry and sad. I understand where it stems from but I still hate us for it.
Will you be in Athens? What job do you do now? Work is tough and if you are not Greek and don't speak Greek you will struggle. Unless you are a specialist in some way.
We are planning a move back to Greece in 2-4 years. I miss it and its flaws and my DH loves the lifestyle there. We won't be in Athens though as we want island life. Currently we have 3 months booked there this summer so we can have our first baby there. I just cannot wait to wander the streets of Athens (it is a good walking city - we walk everywhere) and soak in the sun and the good vibes.
Let me know if you have any other questions. I am happy to help and give you the good, bad and ugly. Also happy to advise on the Greek family and how that might play out for you and your partner. 