I guess you have both passports then. There are a multitude of ways to get both licences when it comes to the order to do things. Flight training is cheaper and faster in the US. Plus there are jobs there. I would suggest you get the FAA 61.75 piggyback certificate thrn a full FAA IR, CPL and Multi add-on, then CFI and CFII. Make getting a job in the US your priority - right now your just need a pulse.
In amongst that training after your FAA IR you can get an EASA SEIR and SECPL if you want to, which will save your ATPL exams from expiry. Then when you get an ATP with a type rating with 500 hours you can do a sim ride for a full ATPL.