I'm assuming 'ATP' was a typo, and you're actually British. If you're studying for your ATPLs, then you already have a PPL and are going modular. This rules out a few big players, the kind that only hire rich 12 year olds, but don't worry there are still plenty of airlines who value life experience. My advice would be to go for the cheapest convenient ATO for the CPL and IR, then spend wisely on the MCC/JOC - this is where you really get to show what you can do in an airline environment. Finding that first job is always going to be the hardest part, but you have to get the licence first. Then you do whatever it takes. Seducing someone from HR works, as does making friends with current pilots. I got hired because I rang up HR and asked if I could send a CV.
You mentioned moving away. 99% of pilot jobs will involve moving, probably too a different country. You can't be fussy. You mentioned ATPL grades. Some airlines will use them to whittle thousands down to a few, and good grades should mean that you will find company ATPL tests easier, but a pass is the most important thing and you have to work with what you've got, you'll forget most of it in a year anyway. Application strategy is what you need. Just doing a mail shot and waiting for the offers to roll in might not be enough.