Robert & Emily’s marriage was registered in Cookstown, Tyrone, 12.1.1865. No children showing as born in Ireland on the Irish civil indexes, so you would expect them to have left soon after that.
There’s a Robert Miligan aged 21, labourer who travelled with E J Miligan aged 20 on the St Mark which arrived New York 30.9.1865 from Liverpool. On Ancestry's summary their nationality is given as American but if you look at the actual passenger list it’s shown as Great Britain. That might well be them. (Liverpool was by far the busiest port of departure for migrants from all across Europe and far more Irish emigrants left from there than ever sailed directly from Ireland). Many of the other names on the passenger list are also clearly Irish, though described as being from Gt Britain.