It would depend on when Michael emigrated to the UK whether or not they'd be included in any existing records in Ireland.
If the family was still in Thurles when the 1901 and 1911 Irish census was done, you can look at the microfilms to obtain information about John and his family in Thurles. The microfilms are available on order through a LDS Family History Centre. There are no census records after 1911. If the children were born in Thurles, they should be included in the census records too.
If his wife died in England then you would need to research those records in the UK. That would be true for any children who emigrated also.
There's no information either in Thurles, Tipperary or elsewhere in Ireland about Irish born who were in the British military. Those records would be in the UK.
To find information about Bridget Nolan in Dublin you would need her parents names and place of birth in Dublin City to find what church register might have her baptismal record.
You can look at the Index of Irish Civil Registrations microfilms to find her specific date of birth in Dublin City in the Indexes. You could also search around the same date for her sbilings births. The microfilms are available on order through a LDS Family History Centre. Some of the actual birth certs are also available on microfilm through LDS. In the event those aren't included, using the information from the Index, you can request copies of the actual birth certs from the General Register Office at http://wwwlgroireland.ie
and click on "apply for cert".
With that information, you can look at the 1901 and 1911 Irish census microfilms for that location in Dublin City to see if the family members are listed and what their information is.