Hello Julianna,
I went to www.rootsireland.ie and searched for Ellen Mills born about 1825. There are 5 matches found. However none of these is in either County Tipperary or County Cavan. You can look at the website to see what counties these were found. You should keep an open mind regarding your information. Tipperary is in County Tipperary and County Cavan is a totally different place further up ;the ;country.
Since you don't have parents names, do you happen to know the given names of her children? Many children were called after their grandparents so this may help you in your search.
As you don't know exactly where she was from, it is worth looking also at www.irishgenealogy.ie to see if you can find her there. Have you searched the ships passenger lists for her? Australia kept excellent records. I know in my own case the informtion gave the names of my g grandfathers parents and where they came from.
I think you will have to do some more digging through the records over there i.e. passenger lists, birth marriage and death records and census records to glean as much info as you can. Only then may we be able to help you with your search on this side of the pond.
Look forward to hearing from you again,
Anne Dennehy