Different doctors will react differently to your issues. Like someone recently said in another thread, there are good and bad doctors in both countries, so really, it's all down to luck.
I don't have cystic breasts, but I do have recurring mastitis in one breast and it's been an issue in both countries. In the US, they prescribed me some antibiotics and sent me home (I didn't have insurance, so of course, they didn't bother to run any tests). Here, I was referred to the breast clinic, but by the time I got in three weeks later, the infection had resolved on the antibiotics and so there wasn't anything to see in the ultrasound. They said it was "a benign problem" and the doctor even said it was somehow caused by smoking.
Well, I quit smoking two months ago and have the infection again. This time, my GP gave me some low dosage antibiotics and the infection got far worse and caused an abscess (she didn't read the notes in my file that the breast clinic sent over stating if I had the infection again, I would need strong antibiotics for four weeks) and so I showed her the abscess, and she gave me some other antibiotics and a week later, it was worse again. I went to a different doctor, who rang up the breast clinic and got me an emergency appointment for a week later and told me to stay on antibiotics until I was seen. My appointment with the breast clinic was supposed to be this morning, but they called and said the nurse I was supposed to see (and I thought I was seeing an actual breast expert or surgeon or something according to the GP who scheduled it) broke her arm and so they shifted my appointment to tomorrow morning. Meanwhile, I've had a swollen, purple, painful boob for nearly three weeks and diarrhea from the antibiotics.
I've not been exactly happy with the treatment I've gotten, BUT, without insurance in the US, I might not have gotten any treatment at all, so I'm trying to maintain perspective.
Basically, in my opinion, since I have had good insurance in the past in the US... my thoughts are that, as far as the care itself goes, good insurance in the US seems to trump the NHS (though they sometimes go overkill and give you some unnecessary tests to squeeze more money out of the insurance carriers) but the NHS is certainly better than living in the US without insurance. Bring your medical records with you, but in my experience, they don't really tend to take much stock in them and prefer to run their own tests based on your symptoms. Good luck, and if you aren't happy with your GP, try to see a different one in your clinic until you find one you really like, just like you would in the US.