Upon overnight reflection I think that the Crowe men who emigrated to Warrington, Cheshire England and Australia and who possibly served in the RN, may have been uncles of Annie Crowe not her brothers. If memory serves me right the uncle who went to Warrington was another Austin Crowe married to a woman called Clare! It was their son who became the mayor of Warrington.
Are there more than two Joseph Crowe? The way I've told it there is grandfather, son and grandson all called Joseph. I do know for sure that Annie had a brother called Joe Crowe and that both her father (our great grandfather) and her brother(s)(your grandfather/ my great uncle) served in WWI.
With regard to Kate Begley, wife of Joseph Crowe and Catherine Begley, mother of William Whelan, these must be two separate women. Catherine Begley of Pollerton Beg must have been born c.1830s or 1840s in order to give birth to my grandfather William Whelan in the 1860s.
Also have you any knowledge of some Crowe women who remained unmarried, working as nurses or matrons?
As my father is now dead and I have no contact with his Whelan relatives I am working from memory. All of Annie's sons eventually moved to England via different life journeys.