Its not going to be easy fennel. Have you checked with airline if you can take buggy right up to plane? Best to check access with airport though as Prestwick involves a lot of stairs but Beauvais is on the level. Usually you can take buggy to bottom of plane stairs and the will put it into the hold from there and bring it back to there for disembarking. Not much use if faced with 2 flights of stairs though.
Your timing is going to have to be amazing to get them all ready to board and to the front when called. If appropriate I would seriously consider pull ups incase of bad timimg, a baby carrier for the smallest so two hands free and the biggest backpack they will allow (don't try two bags or anything else it ain't gonna happen). You will have to really think about what you will need for the flight.
If one of them is big enough fill a small pack pack with little things to amuse them. Go for the novelty of miniture e.g. pencils, colouring books etc but keep it light the last thing you want is to carry this as well.
This is one of the times when sweets are completely necesary!!! Bribe bribe bribe. Not chocolate (very messy) and not to much colours. Can work to your advantage to have a hyperactive toddler covered in chocolate though. They will soon swap seats .
Seriously though if you have timed it wrong and board late people will swap.
Remember you will never see a German or French mum covered in baby food meekly struggling to get through, dragging half of mothercare behind her. No she will march to the front with her immaculate children and a spare nappy and wipes in her designer matching hand bag .
(no offense meant to any immaculate european mums, were just jealous).