Don't carry the toddler but get a good quality baby carrier. How old will s/he be? I have a bad back and can't constantly lift so my ring sling was a literal back-saver.

I've done the transatlantic, with connection, alone with three small tots many times.

If the toddler has a seat, you can use the car seat-on cart apparatus instead of a stroller. That would mean they won't take it away from you at the gate so no worries there. I usually borrowed a stroller at my destination. Carry-on's can be hung on it.

Take no more than two carry-on's. Get rid of your purse and stuff it in an oversized backpack with what you really need for the flight. I also would wear a string bag with just the passports, cell phone and wallet. Then either a second oversized backpack or a duffle with all your back-up supplies like extra diapers. That will be stuffed in the overhead while the backpack remains at your feet. No luggage with wheels! Perhaps for the 5 year olds if theirs' are small.

Everything else get checked.

The two 5 year olds can carry their own backpacks with whatever they are bringing for entertainment. You might want to take any tablets or anything else of real value in your own bag.

What is involved in the transfer at LHR? Do you have to change terminals?