OK, here's what a normal person would have done
I mean -- a normal young woman in 1950 with husband she just MARRIED in America. She would have gone back to Ireland (hey folks, they did have airplanes even back then) -- wore her wedding ring, and told everyone of her wonderful luck in getting an American husband! -- packed up her widowed mother -- brought mom back to the US -- and all lived happily ever after, with Tony in their new house on Long Island.
Problem solved. Movie -- 20 minutes long.
*****
BTW, I actually knew a couple -- my best friend's grandparents! -- who married secretly. She was of Scots ancestry, and he was an immigrant from Italy with a heavy accent. Their parents were not thrilled with them even dating. They met at college in the 1920s. They married secretly, just like Eilis and Tony, and hid it from family for about 2-3 years!
Why bother with this? They wanted to sleep together. And back then, nice people did not have pre-marital sex.
Also this was a "mixed religion marriage" -- she was Presbyterian, he was Catholic. That sounds like nothing today, but it was real intermarriage in 1924. People were upset.
Later, after graduation -- they got "pretend engaged" and had a full, formal church wedding. Parents accepted it. Kids were raised Presbyterian. They did not tell their children or grandchildren until the 1970s about the secret marriage!