Oh yes, I’ve done it lots of times, starting from when my youngest was very, very tiny. It’s absolutely fine, but you have to be organised.
If you take a buggy through the airport you need to ensure that 1)you haven’t filled it full of stuff that you then have to remove at the gate and 2) that you don’t have anything else in your hands at the gate except a walking child’s hand and the folded pushchair.
Most airports are stairs down to the tar Mac so you won’t be able to push the pushchair right out to the plane.
You will need a sling/carrier for the baby.
You need to have the toddler well schooled in advance that they must sit down with the seat belt on. Practice this and tell them plenty of times in the lead up.
Put the toddler by the window 1)so they can’t escape and 2) so they can try and find your house and wave to daddy (or whoever) as you go up.
Have some toys/stickers whatever. Don’t take them all out at one time. It’s worth packing additional stuff in case of a delay. We have a fold up black board from Crayola I think as well as a travel aqua doodle thingy. Good for the actual airport.
On the plane, have a small bag (I used to use the small M&S bags for life, the little teeny one), with snacks, wipes and a nappy for each, so it’s immediately to hand. Keep it down the side of your seat. Don’t put anything on the floor, you won’t be able to reach it.
Have a nappy bag or two to hand for rubbish. (Wipes, snack wrappers etc, it’s easier than chasing it all down)
Use bulldog clips to clip pages to the table for colouring.
Make sure both babies have had a change of nappies immediately before the flight, and you do a pee.
Have a small change of clothes for everyone (you too).
Know where absolutely everything is in your bag. It really makes things easier.
At the airport, if you have a push chair, it will have to be separately screened. Your toddler will also have to be separately screened. Prepare them for this...!
Oh also at the airport, take a zipped hoody or whatever with preferably zippable pockets, for you to put your watch,coins, phone, whatever else in for going through security so they aren’t floating around the bottom of the container.
Ask the person at security not to put your stuff through until you are ready. I’ve had a few times where the iPad is through and I’m not, and it’s too much to worry about.
People are generally helpful. If someone offers accept it!