How did they get there?

[Andrew Tarr]

I would also favour Bristol. Railway reached Anglesey in 1848. Was Holyhead an option?
Irish Genealogy Toolkit has pages on emigration. The section on Britain has a bit about migration routes.

Thanks for that, M-S.  Holyhead would have been an option, but a long (and expensive) journey, and unlikely for people heading (I assume they knew roughly where they were going) for southern Ireland.

I've had a look at the Toolkit page, but unsurprisingly it concentrates on those leaving Ireland, not immigrants, and it doesn't mention any southern ports.  I'll look again tomorrow.

[Maiden Stone]

Having arrived in Ireland, depending on which port they landed, they might have travelled onward by train. The first railway line in Ireland opened 1834 between Dublin and Kingstown (Dun Laoghaire) as part of the Royal Mail route between London and Dublin.
The Great Southern & Western Railway from Dublin to Cork reached Cork in 1849.
There were several other railway lines connecting most regions of Ireland by mid 1850s.

Middleton was a post town. It's only 10 miles from Cork City, so hardly in the middle of nowhere. Investigating the Royal Mail routes in Ireland, England and Wales may be fruitful. There's a book "History of the Mail Routes to Ireland to 1850" by George Ayres. Preferred route was Holyhead to Kingstown (Dun Laoghaire). So if your family could have got to Holyhead, onward travel should have been efficient, although they may have had to change trains.
Articles about the history of Irish transport 18th & 19th centuries:
"Ireland's time-space revolution" History Ireland website.
"Charles Bianconi and the Transport Revolution 1800-1875" The Irish Story website.
 
County Cork was well-supplied with army barracks. Rapid movement was vital. (1798 Rising and French invasion emphasised that.) Royal Engineers were stationed on one of the islands at Queenstown (Cobh). They surveyed and mapped Ireland in detail during first half of 19thC.

[Maiden Stone]

Cork Ancestors website has, among other gems, some extracts from "Cork Constitution" newspaper.
www.corkgen.org
1826 editions had adverts for steam packets to Bristol and Liverpool. Lists of arrivals & cargoes at "Cove of Cork" included coals from Milford and slate from Bangor and many colliers with coals, departure port not named.
1896 edition advertised sailings from Cork to Cardiff, Milford, Newport, Plymouth and Southampton. I found the 1896 edition by looking under heading "Shipping".
 I don't know if there are extracts from 1850s editions of the newspaper.
Other items I found were lists of deaths on emigrant ships from Cork in 1847 and discharged soldiers who were natives of a town. There are bits about many Cork towns. Illustrated.