Hi
BMI does not fly Trans-Atlantic.
The BMI ticket will be codeshare on United, so you will have a United experience on a 777 in Business.
Given virtually all United 777s on international routes still have the old cradle seats, you can decide for yourself whether the cheaper fare compared to BA - which has fully lie flat seats - is justified. I would suggest the BMI ticket is cheap because United's long haul business class product on the route is significantly inferior to BA.
regards