I'm not aware of a passport check-line dedicated to first class fliers, but you will exit the plane first and thus be ahead of a few hundred others. Last year I pre-purchased my coach ticket from LHR to Bath and saved on the fare; if you don't have a reservation in advance you may not get on the first bus. They fill up with people with reservations. Passport control can take anywhere from a few minutes to over an hour, so it's difficult to know what time you will get to the bus station (which is a 10-15 min hike from passport control). I would allow a couple of hours from the time your plane is scheduled to land for making a reservation. My plane was supposed to land at 7:10 am but actually arrived at the gate closer to 7:30 am. I exited passport control at 8:40 and hiked on over to the bus station. The next bus (coach) to Bath, and the one I had a reservation for, left at about 10:15. I arrived in Bath at about 1 pm completely exhausted. And this is why I will no longer take the bus from LHR to Bath. From now on, I will go into London first and spend at least one night before traveling to Bath; and I intend to take the train.