Hi,
Robert and Edith were definitely estranged by by 1895. Robert and Edith had their last child together in 1894 and by July 1895 she was with Levi Brierley.
On the birth certificate for Archie Brierley, the name of the father is blank but Levi Brierley is the informant. If Archie was Robert's son, I think it very unlikely he would have been given the middle name of Brierley. Edith styles herself as Edith Veevers formerly Osborne. In 1896 when their next child is born, Levi Brierley is named as the father and Edith is now Edith Jenkins Brierley formerly Osborne.
I believe Levi and Edith didn't marry because a marriage would have been bigamous. By the time Robert and Mary Ann married in 1903 and Edith and Albert Pollard married also in 1903, Robert and Edith had been separated for more than 7 years. So although the marriages were illegal, they were not bigamous. Under the 7 year rule, if Robert and Edith had not seen each other and believed each other to be dead, they could remarry without the fear of prosecution. I have not found any evidence of divorce and believe divorce was uncommon at this time among the lower classes due to its expense.
On their marriage certificate dated 28 March 1903, both Robert and Mary Ann said they were widowed. Robert wasn't widower as Edith was still living. Not sure why Mary Ann said she was a widow though. Albert Pollard and Edith Osborne were married on 24 January 1903. Albert stated he was a widower which is correct but Edith said she was a spinster which she wasn't.
With regard to the reference to Texas, this is still a mystery. Mary Ann's father, Eli, was born in Preston and her mother, Mary Ann, was born in Liverpool according to the UK census. I haven't researched her parents any further back.
I think I might have found shipping information for their trip to New Zealand but need to look at this more detail so will get back to you on this.
Luzzu