Of course you would feel the cut. I didn't have a paycut when I moved here and I felt a massive difference in disposable income (granted, I didn't come from NYC and had a much more reasonable cost of living in the US). You do adjust though.
I'm lucky. My parents have always committed to me to visit annually and I've committed to them that I'll visit annually. We've actually done much better than this in reality. But my case is rather rare. I know many expats who are not fortunate enough to receive regular visits from their parents. And their budgets simply don't allow them to return to the US. I am grateful for what I have and am able to afford - no doubt.
I leave for my annual trip to the US tomorrow - woo hoo! I'm headed to Clearwater, which is where my parents live and I grew up. But today is my brother's 40th birthday (he's in Boulder). My parents flew in to surprise him and there is a big surprise party tonight. I got a LOT of pressure to go. But it would have added £2,000 to the cost of our trip. Plus a 7 hour time difference with a 2 year old is no joke. Not to mention a car and hotel (which we fortunately don't need at my parents). The cost was just too high. Yet, I'm bummed to be missing these events!
One thing I have found living overseas - everyone thinks because you live overseas you must be LOADED. While you will certainly meet your fair share of expats in London who are VERY well-off - most are just making do. No matter where you live, it's all about budgeting and living within your means. I personally believe that living in both the US and UK have their pros and cons. I do not believe one is superior than the other. It is truly individual preference. For me, it's quite equal. Though maybe the US beats the UK a bit.
A very accurate website for seeing what your take home pay will be is this one. Could help with your discussions. http://www.thesalarycalculator.co.uk/