Please do not just assume that Easyjet will be cheaper than Flybe. Sometimes it is and sometimes it isn't.

Easyjet and Flybe both have perfectly acceptable services to London. I guess you will want to look at price, schedule and your choice of LTN or LGW.

You don't get a seat assignment on Easyjet but you can pay extra to board first. Otherwise you will get to board in the second group as you have an infant. On Flybe you will get assigned seats but if you want to choose them and ensure you sit together you will have to pay extra.

You can book the Flybe flights as BA on the BA website if you want - the price may be different and if it is it will probably be higher.

It's a long train journey. You can do it by day or overnight. The overnight service is primarily a sleeper service but there is also a seated option (in my view extremely unattractive). Sleeper cabins have one or two narrow berths. I'm not sure that would work well for a couple with a young child.