I have begun to explore a new option. (for me in anycase)
We know that many Irish people began their journey to America and elsewhere via Liverpool. Some of them stayed in Liverpool, at least for a time.
The 1851 Census for Liverpool (England), via ancestry, gives a total of 67,291 Irish born people living in Liverpool. The majority give no other birthplace but Ireland. However there are exceptions.
For example in Paul Street, Liverpool, James Brennan, born about 1796, is recorded as having been born in Mayo, Ireland. James was a lodger in the house of Thomas King born about 1824 in Clare, Ireland. Thomas King's wife Irene was also born in Clare. Ann & John Portill, also born in Clare and described as relatives of Thomas King are also resident in the household. Of the 10 people living in the household, only one was born in England. In fact there are many Irish families resident in the area.
I highly recommend a search for your missing family in the Liverpool Census of 1851 and 1861.