Skipping the line at Heathrow?

When you transfer in Heathrow you don't have to go through passport control/immigration however, you do still have to go through security control. You go through customs and have passports checked when you arrive at Edinburgh.

As for how this all plays out after Brexit, it's a case of no one knowing at the moment. Best guess is be flexible.

Sorry, Heather, the poster is flying from outside the UK (they don't say that, but I assume so) --> Heathrow --> Edinburgh. In that case they will go through immigration (passport control) at Heathrow, as that is where they enter the UK.
As this is one ticket (Flybe code-share), their luggage will be transferred and they will get it back in Edinburgh, which is where customs (a technicality) takes place.

As far as risks are concerned:

• Brexit, should not affect it, as you are not going via the Schengen Area. Small risk the queues at Heathrow are longer as EU citizens will not be going through the EU lanes. Not totally sure, as it depends how they re-work immigration
• Flybe has a small risk factor, a number of airlines have gone bust recently, Flybe has just got a cash injection, so risk is small.

When flying Business Class on British Air or having Sapphire level on Oneworld we got fast pass through LHR immigration. Also US citizens can obtain the British version of Global Entry but need to have visited the UK at least four times in the past year. We have received info on this program by email after visiting LHR.

... the poster is flying from outside the UK (they don't say that, but I assume so) --> Heathrow --> Edinburgh. In that case they will go through immigration (passport control) at Heathrow, as that is where they enter the UK.

Correct, we're flying Chicago-Heathrow-Edinburgh. I expected that Heathrow would be our UK entry point and would go through immigration there. I've just heard suggestions that the lines could be longer after Brexit and wondered about this "skip the line" option and whether it's in effect now.

To be clear, I'm not inordinately worried about Brexit messing up our connection and, even if it does, Flybe has multiple flights to Edinburgh, so we'd just catch a later one if we're delayed by long passport/security lines.

What you are referring to is allowing U.S. citizens to use the e-gates when arriving in the UK. It is supposed to start this summer but no date has been announced.

The other program is called "Registered Traveler." It also allows you to use the egates. I have it. To qualify you need to have entered the UK four times in the previous 24 months. It costs 60 pounds/year.

EDIT--changed hours to months.

... you need to have entered the UK four times in the previous 24 hours

Well, that sounds ambitious. ;-)

Seriously, thanks for the info, Frank. The effective date is what I was unclear on. Several of the stories I found said mid-2019, while a couple suggested January.

When I arrived in Heathrow in December, I didn't find the lines inordinately long. I did see a fair amount of advertising for the Foreign Traveler option, but requirements and cost made it ineffective for me. The eline wasn't so much faster to make a difference - that I saw on that given date.