41st foot. Stationed in Ireland 1800's

[Maiden Stone]

Mullingar is where my gg grandfather signed his attestation papers for the Crimea war, Is that possibly near Athlone Barracks?

Distance between Mullingar and Athlone is approximately 25 miles.
Railway line between the 2 towns opened 1851.

[Maiden Stone]

My gg grandfather was in the 41st Welch Regiment of Foot, which was renamed the 41st Welsh Regiment of Foot and THEN renamed with different numbers.

My advice is to look at the 41st Welch Regiment of Foot, which I think changes names to something like the 87th, but you will be able to follow where they where and which battles they were involved in. 

I would look at the archives online in the UK, NOT in Ireland, the services fought under the UK banner, so the records are held in England too, they are online. 

National Army Museum. 41st (Welch) Regiment of Foot. Brief history. Amalgamated into The Welsh Regiment in 1881.
https://www.nam.ac.uk/explore/41st-welch-regiment-foot

The National Archives. Research guides.
"How to look for records of British Army soldiers up to 1913"
https://www.nationalarchives.gov.uk/help-with-your-research/research-guides/british-army-soldiers-up-to-1913/
There was a joint army establishment from the Act of Union 1800.

[Maiden Stone]

I'm trying to find information on one of my ancestors. His name was Peter Wright and he was a soldier in the  41st foot, stationed in Ireland.
I know he was in Dublin in 1856 as that is when my 2nd great grandmother Drusilla Erina Emily Elizabeth Wright was born.

Birth of a man's baby isn't evidence that the man was there at the time. His presence wasn't necessary.