Generally you don't have a right to work 50% or any part time in fact. As a carer you have a right to 'request' part time working. Having said that if a company already has people working part time they are on a very sticky wicket if they turn down your request and not somebody elses.
Specifically for BA I believe they have 2 part time rosters essentialy month on, month off and 2 months on 1 month off (don't quote me on this - this is just what I've picked up on 'galley FM'). I the month off I blieve you have to do one duty to maintain recency. They also have 2 part time 'schemes'. One for people who have a legal right to request part time working (i.e parents, carers etc) and one for 'aspirational' part time, for people who want to but don't have a legal right to request it.
I very much doubt you can do it from day 1. The training is pretty intense - my feet didn't touch the ground for the first month, and then the line training is rostered for you after that.
I've got no idea how you bid for part time, but i suspect its on an annual basis along with the fleet move bids, command bids and the like.
I worked 75% part time in my previous airline as its the only way I was ever going to get a life and see my kids. I'm more than happy full time at BA for the moment (a full time BA roster has more days off than my 75% 'cap371' roster did at my previous airline!).
I don't know where you work now and under what conditions, but full time, but with some control over your roster, and 100% roster stability is a huge change in lifestyle and home life if you are used to alot of disruption, and you may find that this gives you the home life you are looking for without looking for part time too.