I make the United-BMI LHR connection quite often. They are both in Terminal 1 which helps. United will check your luggage through. BMI is a Star Alliance member so you should be able to use the lounge assuming your level is high enough. They stopped announcing flights in the lounge so make sure you pay attention to the monitors
You will have to go through customs at LHR. Make sure you get into the 'Connecting Flights to UK/Ireland' queue (to the left side and around the corner) and not the general arrival in UK immigration line.
You will have to answer the standard immigration questions (make sure you have the address where you will be staying. Not having an address is a red flag) you then have your picture taken and will be given a card. Then you will go through another security scan/carry-on check. They will not allow ANY liquids through.
Finally, you will come out into the main terminal. I usually stop by the BMI desk (or lounge)to see if my bags made it through (have your luggage tags out) and check to see if there is a gate assigned for the flight yet. Once the gate is assigned (40 minutes before the flight) you'll have to go through another security check where they will check your picture/passports and then a (potentially very long) walk to the BMI gate.
In January we did it with an 1:45 and a 1 hour layover. We actually had plenty of time to spare but a single delay could cause a major problem so I would suggest at least 1:30 to 2 hours.