I'd definitely have to take a pay cut. I'm paid very well in the US - definitely on the high end of the range for my current job. And pay in the UK is on average lower, so I don't think they can match it. But my wife and I really want to travel within Europe as well as experience living overseas while we're young enough to enjoy it.
When I moved here 7 years ago, I moved for a job (I am also an engineer) and also took a huge pay cut. I'm paid very well for the area I live in, but I'm still making less in equivalent US dollars than I did when I got my first proper engineering job. Was it worth it to move the UK?
Hell yeah! I didn't come for the money, for sure. I came for the experience, the excitement, the adventure. Now, I'm British, I met my other half, and it's home.
Also, despite the pay cut, in seven years, I was able to pay off $18,000 in US debt, all my student loans, develop a nice pot of savings, and pay into a decent UK pension pot (nothing like my 401k, but at least that's sitting there nicely growing slowly). I've also been able to take holidays, never have to worry about health care (yay NHS!), drive a nice car (we only have one in our household), and eat good food. I'm currently building a nice eco friendly house in the coutntryside with my partner as well. I've cut back on all the extra spending on things I don't need, I grow a lot of my own food and make my own wine/beer. Having a paycut forced me to look at the important stuff and budget accordingly and I now live within my means. (My $18,000 of debt came from a time when I was not living within my means, a foolish young girl I was)
I'm incredibly fortunate and lucky, but I will never take this for granted, as I grew up quite poor and worked very hard, pushing my way through engineering school whilst working three jobs to get where I did. And you never know when life circumstances will change.
So my long winded point is, if there is an opportunity to move, I say go for it!!! You will find the UK is cheaper in a lot of areas (I couldn't believe how expensive it was food shopping in the US last time I was there) and the NHS comes from your taxes - so no extra health insurance premiums. I also find things like car insurance cheaper.
Good luck in your journey!!!