I can't comment on cost as I was a resident and so went the state route, but I had my twins in Greece 12 years ago. Stayed in a state hospital for 1 month as they wanted to come early. Food pretty decent. Ward with about 4 or 5 beds, I saw patients who'd had hysterectomies and miscarriages go through the same ward. No one really spoke English (my Greek was reasonable but not with medical vocab though). A little TV that was on constantly to Greek soaps. Staff pretty lovely. A lot of Greeks had travelled to Crete specifically to have my consultant as he was very good.
Antenatal care fine, regular appointments etc.
Had a C-section as soon as I hit 8 months, the consultant whipped my upstairs and within about an hour they were out. Epidural fine, no choice for partner to come in but he didn't want to. Babies were healthy but small, the smaller one was transferred to capital city hospital as no NICU where we were. My mother fought for him to stay one night with me first. After care...you absolutely had to have someone with you overnight. I had a catheter for a day, but then my mother and partner took turns to stay with me, on a chair. to help to toilet etc, no nurse help available, and to care for baby. Nurses were good showing my husband what to do, but I don't remember them helping after that. No help with breastfeeding. I had a panic attack when first trying to move after C-section, and again when leaving the hospital after being inside for a while, and they were nice, very matter of fact, but not the 'extra' compassion I feel NHS can provide. All injections etc for babies had to be paid for, and because it's common not to take babies out for 40 days, we paid for paediatrician to come to home to do the first ones.
Don't worry about nationality, they automatically qualify as British citizens through parent's nationality.
Recently had two East European employees give birth in the north of the NHS within a week of each other. One had a complicated pregnancy ending in C-section, the other a straight-forward very large first baby. Both had excellent care, both released after 2 days, very friendly staff.
I wouldn't travel back to Greece if I had more children, although I was very happy with my care, the NHS is amazing here, i think you just have to be vocal about what you would like and do your research.