I flew from NI to UK on my own when C was 6 weeks old, went with flybe. I had buggy base, maxi cosi and change bag. I kept him in buggy until scanner part. I then had to put maxi cosi and change bag through, while carrying Colby. The buggy base was taken by staff to be checked elsewhere. I had to take my shoes off so did that before carrying him through. Staff offered to help (but I'm stubborn and wanted to prove that I could do it myself). I also had coat etc draped over buggy so easily removed to go straight through check. Then as soon as through popped him back in maxi cosi. He stayed in there right up until departure gate door, where I explained to staff how to collapse my buggy while I held C again. My change bag was big enough for magazine, expressed bottle etc (didn't have to check milk) they just asked to look through bag as I had left my lansinoh cream in it be accident, they took it out and put in my plastic bag. Only part I found a faff was then getting my things gathered together etc
I was in a window seat beside propeller, I fed before we took off, he slept whole way across and then fed after we landed. I think the noise and vibrations of propeller put him to sleep and thankfully I had a really nice guy beside we who didn't mind that I had baby etc.
I'm sure you'll be grand and there is plenty of help along the way especially when they see you're on your own. Have a lovely trip back to see your parents