When Amy and Rory get stuck in New York, they try to leave at first. I mean, of course they do, they’re not idiots. If the Doctor can’t come here, why can’t they just get out of the center of the paradox? And even if he still can’t pick them up….well, no reason not to see some of the world by the slow path.
It’s just not that simple. As it turns out, the contortions of time and the way it’s adhered to them trap them physically, not just temporally. When they try to leave NYC, time goes…weird. They find themselves back where they started, or time slows to a crawl, or starts skipping and fracturing for them. They wait for trains that never arrive, or drive all day and get nowhere, or walk and find their strength sapped away. You get the picture. So…they stay. They poke their boundaries every so often, but mostly they just stay.
It’s not so bad. If you had to pick one place to be stuck in for the rest of your lives, New York City from the 1930s onward certainly won’t be boring. And River can visit, occasionally, even if the Doctor can’t. (Sometimes she leaves them gadgets to keep them equipped for any timey-wiminess that may arise, like the thing that looks like an egg timer and detects artron anergy within the city confines. They always wonder if she has a Reason for these things, but they don’t ask.)
They build their lives. Rory stays in medicine, of course, even if being a male nurse is a little more unusual in this period. Amy writes, fantasy stories about little lost girls and brave boys and strange wizards with funny blue houses.
(In her stories, children always find their way home in the end.)
After WWII they adopt a little war-orphan baby named Anthony. He, more than anything, anchors them. Life goes on. They settle, like the foundations of a house.
Until one day in the 60s when Amy (older, but not slowed down yet) bursts into the house, grabs Rory, and says “She’s going to be HERE.”
“1969. The Moon.” Amy waves a copy of a magazine with headlines about the space program at him. “MELODY.”