This is a story which interests me because I'm a member of BMI's frequent flyer program, and I've used it to amass almost 150,000 miles, which I'm going to use at some point for flights to London. Their miles do not expire, and can be redeemed on Virgin Atlantic as well as BMI itself.
However, my experience in the US is that unless you have Gold status with BMI (equating to Star Alliance Gold status), you don't get lounge access with United. Also, the miles which I amassed with BMI were useful for transatlantic flights, but not so much inside the US. For example, i could not do much with a small-ish amount of BMI miles, though if I had miles from a US carrier then I could use a small-ish amount of miles for internal US flights.
Therefore, a few years ago I signed up for United's frequent flyer program and I use this to amass miles (which I can spend internally in the US, or for transatlantic flights, e.g. with Aer Lingus to Ireland). I find that I get perks such as upgrades, though still not lounge access (as I'm still only on Star Alliance Silver level).
Here is the story from Yahoo Finance anyway, even though it doesn't match my experience with lounge access (unless you have Gold status):
BMI is part of the Star Alliance, a group of 25 airlines that work together (United, Lufthansa, Air Canada and Air New Zealand among them), which means your points and upgrades can be applied to any of those airlines. While most U.S. airlines do not offer complimentary lounge access to frequent fliers, most foreign carriers do, which means you get the benefit of using those facilities within the U.S.--on airlines like United--without actually belonging to their frequent-flier programs.http://finance.yahoo.com/family-home/article/106661/How-to-Score-Secret-Airline-Perks