LHR: Designed by Satan himself?

Which terminals are you flying in to and out of? This should be printed somewhere on your itinerary. There are underground walkways that connect all of the terminals--meaning that you shouldn't have to walk outside!--and they're generally quite well sign posted. When you get off the plane, follow the connections signs and then the signs to the appropriate terminal.

Jesus, the arrival roads? Mad. No, all five terminals at LHR are connected by the Underground. Terminals 1, 2 and 3 radiate off a central hub, while Terminals 4 & 5 have their own Underground stations. If you're transferring between any any of terminals 1, 2 or 3, you can simply follow the signs to the right terminal. Walking slowly, this will take you roughly 20 minutes. As Jakey said, if it's BMI > United, your bags should be checked through for you. If in doubt, ask a BMI person to help you as you get off the flight, it's their entire job to make sure you get to your connection. If it all goes a bit wrong and you find out you have to get from Terminal 1 to Terminal 5 or something odd like that (note, this is very unlikely as different airlines fly from different terminals, so the only reason they'd change it is if your plane broke down and they stuck you in another flight with another airline), don't panic, and ask an airport staff member to take you to the inter-terminal buses. Or go downstairs to the underground/railway station and ask a staff member to help you get to the right terminal - the underground and surface rail trains loop at the end, so there is usually free interchange to the other terminals. Good luck, and don't panic!

Thanks folks. Jakey, I fear what you're saying may be too good to be true, but I just dug through the Heathrow site for awhile and you're right, it does seem to say I'll both arrive and depart from terminal 1. If that's correct, I'm sorry to have wasted all of your time! That said, thanks also to Happy Dave for the next time I'm stuck going through LHR or if they DON'T arrive & depart from the same terminal. I seem to recall last time walking partway underground and partway along the arrival roads (also dodging pits filled with boiling oil, attack leopards, and poison arrows) but as I say, I was lost - For all I know I went full circle around the entire airport and returned back at the terminal I left (ok, not really, but still). Also, I think you mean "Keep Calm and Carry On". :) Thanks!

Failing that there are shuttle buses and even the tube line connecting the terminals. 1,2 and 3 share a tube station, 4 and 5 have their own stop.

If you're really late tell the security personel that you are going to miss your plane and show your ticket, they'll make you go to a different, much quicker line.

(You know we're having a http://metatalk.metafilter.com/16623/San-Francisco-Meetup on the 29th, right? If you're here and available, join us!)

you'll be OK

for what's its worth, T1 is relative delight at the minute. since BA moved to T5 it's operating at partial capacity, and the handful of times I've been through it in the past few months have been pretty pleasant. YMMV of course.

It's certainly going to be easier connecting from the short flight to the long-haul one, as most of the delays in Heathrow (at least in my experience) are when a number of long-haul flights come in at once and cause gridlock at security. That shouldn't happen when you come in from Dublin; in fact, as others have commented above, you should be able to head straight towards the normal check-in security clearance as soon as you get off the plane.