Is going through customs with carry on bag faster?

If you only have carry-on, you won't have to go through either passport control or customs in London. Just security. You stay airside and just transfer from one terminal to another via bus. This website tells you exactly what you need to do: http://bit.ly/bKWCHl You will go through both passport control and customs in Lisbon.

Short answer, no. Longer answer is still "no", but here's some explanations and clarifications. "Customs" is not the same as "immigration". Usually, unless you have something to , you walk through the Customs line without stopping. Immigration control is where the passport is stamped. This is the first control point after you leave the plane and it usually takes the longest time. Whether you have your bag with you or you pick it up later at baggage claim makes no difference in your waiting time. Second, you won't go through any of this in Heathrow anyway because the UK is not your final destination. There is a security check, which can form quite a long line. If anything, carrying all your baggage with you may slow you down by a minute or so. Third, two hours is enough time to change planes, but not by much. By the time you get off the plane, take the bus to your connecting terminal, and go through the security check, if your plane landed on time, you will probably have significantly less than 1 hour until it is time to move to your gate (you don't find out which gate until shortly before it is time to board). So, save the reunion for another trip.

Plus, to see your friend, you will have to leave the "transit" zone and seek entrance into the UK by passing through immigration. She will not be able to enter where you shall be and you wil have to exit to where she shall be. 4 hours would have been fine, but 2 hours is impossible. Your friend is right.

Oops, I just realized that I didn't answer your full question. (Serves me right for trying to answer in the middle of the night my time.) I agree with the others, you won't have enough time to meet your friend. Heathrow estimates your minimum transfer time is 75 minutes IF you don't go through passport control and customs. If you do that, two hours won't be enough time. I would also suggest carry-on for a big reason: three flights. The more you change planes, the better odds of a bag being lost.

I wholeheartedly agree with Frank to carry on your bag because that way there's almost zero chance of losing it before you get to Madeira. Your plane to London might be late and even if you make your flight to Lisbon, your checked bag might not. I would worry more about losing my checked luggage than how long it takes to get through the airport.

OK, thanks Frank & everyone else. Carry bag, hustle for the bus to next terminal & if I'm lucky, talk to my friend on the phone!
Even though this trip isn't until mid- June, it helps to ask questions as they come up. Thanks for the swift replies!

It seems faster to me for two reasons. First, you don't have to stop to pick up your bag. Depending on flight schedules that may put you first in line at customs and you can sail through. If you get caught behind a big line of other arrivals, though, you'll still wait. Second, I never get inspected. Maybe I look honest but it's got to have something to do with my one (half-full) carry-on and no second item. I think inspectors are more likely to check people with lots of luggage. One inspector told me about a newbie sending Jerry Lewis to secondary because he had multiple steamer trunks for just one person (the inspector didn't know who Jerry Lewis was).