if I were you and you want to make it happen, get the ATPL theory exams done/out of way, as it'd take while and get it done while you're in your day job. European stuff, obviously. NZ licenses are good for starting the distance course. ONLY THEN do the flying bits with sufficient funds at hand - frequency and intensity would help considering your training took long time ago if you haven't flown since.
Word of warning. The 'low timer' jobs in Europe are mostly instructing or co-piloting offshore (with going straight to utility in Scandinavia doable, but NOT fitting your family life for sure and it'd be ground crewing journey first, too) and the euro IR is pretty darn expensive to self-fund. I'm not promoting it, it's akin to airliner self funded type rating without company paying for it, but that's the sad state of affairs we live in.
Your US visa options other than M1 for training or college/Uni F1 visas (not the 2yr 'ex-J1') are more limited since you've got CPL. If you want family break, go for the (pricey) Bristow modular or shorter CPL etc and Euro FI rating there or someplace in Europe. Again, have the theory all passed before sorting visas/tickets. I've got to know one Scandinavian pilot who did his training ages ago in Australia, then dormant for 7 or so years, like yourself, but he's only just about getting there with offshore job in Europe and he spent plenty cash in US and Sweden on various training/ratings.
In helicopter world, it's not as simple as rocking up with FI certificate/rating at any school and getting job, as in fixed wing world. It's lower cost, lower 'risk' but also low income job to start off, so people better off simply burn tons of cash, seeing the ads asking for ME IR(H) flying to rigs. Presuming you've got 150hrs+ helicopter, the specific types of hours, training, paperwork, that'd be some 15-30k EUR and if doing the extra hours and FI course, double that, or if 'just' MEIR(H), then some 40-55kEUR if you don't have valid ICAO IR(H) or EASA IR(A) for some credit.
As you're not new to the game, consider job market, costs, time 'to get there' etc. Yes, it may not be very family friendly, whichever way initially. Helicopter servicing history, even if not substantial, would be beneficial in some areas of helicopter ops, far from elsewhere utility, for example.
Work rights not an issue in EEA/EU, other stuff covered a lot if you use the sticky or search for things. Just my little chipping in about things you touched on.