As you appear to be booked on one ticket rather than Houston to Heathrow & Heathrow to Cyprus as separate tickets then all you do is follow the signs to "Flight Connections", act dumb and let them point you in the right direction.

http://www.heathrowairport.com/heathrow-airport-guide/flight-connections

That your friend doesn't even know that outside the US connections rarely require you to collect & recheck luggage, let alone have to pass through immigration & customs, perhaps you should listen a bit less to their advice

If I miss my flight to Cyprus, there isn't another one for a couple days, so I miss my meeting(s).

There are 4 flights a day from Heathrow to Larnaca, Aegean fly daily to Laranca. If Things Do Go Wrong then there are two later CyprusAir flights to Laranca. At worst you should be able to get the following day's Aegean flight.

But then missing the flight is always a risk if making a connection so you should have taken that into account before booking even if it meant arrving a day early.

That makes me feel a bit better, but I'll need to check w/the travel agent today to see if it is indeed considered one ticket. I fear it isn't because of the pricing (would have been more expensive that way). Otherwise, I'm good at acting dumb and letting folks point me in the right direction.

Thanks.

It's very possible to fly on two different airlines on a single ticket. I suspect you're thinking it would have been more expensive to continue from Heathrow on "United" all the way to Cyprus. If on a single ticket, you'll be able to check your bag through, and the connection is quite possible. It does take some time (mostly for security at Heathrow), but this connection can be made.

Both United and Aegean are Star Alliance members so I suspect it's all on one ticket and there should be no worries in making the connection. Bags should be tagged through; in fact even if they are separate tickets (unlikely) the UA person at Houston ought to be able to tag the bags all the way, provided you give them full details on the Aegean flight.

Don't listen to your friend on this one! He has you needlessly worried and confused. It doesn't take 'forever' to get between terminals at Heathrow, at least no more 'forever' than transferring at any other large airport in the US or overseas.

When you arrive at Heathrow, just follow the signs for Flight Connections, as noted above. You will eventually arrive at a coach that will drive you across the tarmac of the airport from T4 to T1. You won't go through UK immigration controls and you won't leave the 'secure,' 'airside' part of the airport.

You will most likely go through a security check, either when you pass through Flight Connections in T4 or when you arrive at T1, though I'm not sure which - it's been several years since I've flown into or out of either T4 or T1. If you are running late for your connecting flight, it's perfectly permissible to approach a staff person, tell them that your flight is departing soon, and ask to be escorted to the front of the queue.

Terminals at Heathrow are large, you may land at a gate which is a 10-to-15-minute-or-more walk from Flight Connections, so don't dawdle.

At Heathrow now and just transferred from T5, where I arrived, to T3, where my onward flight departs, so thought I would add a comment about timing.

Arrived at T5 gate at 11:50. Ten minute walk to Flight Connections, where I used the Fast Track lane through security. I was travelling in Club World so could use FT anyway, but there was no checking of boarding passes, just a smiling young man waving me through.

By 12:05 I was on the coach, the doors closed and we set off for T3. Arrived T3 at 12:15. At T3 there was no additional security check.

This is plenty of time. Don't dilly-dally, but otherwise it will be fine.