There is a report in the Christchurch Times 25 Apr 1874, regarding her elopement with William Purcell just prior to the article. While Bartolomei had been at sea shortly after their marriage, it seems his wife had spent £30 and obtained £18 on a dock bond, withhout his consent. William Purcell (also spelled Parcell in some censuses), was a commission agent living at Hope Place Liverpool. He had it seemed borrowed money from her and she had also been supporting his mother and his sisters. He had also written to her wanting to elope with her and had spent more time with him than her husband in a period of 3 months.In spite of Bartolemi having forbidden them from seeing each other, Parcell visited in April 1874 and they ran off together, taking various moneys, coins and jewellry belonging to Bartolomei's daughter who was being educated in a convent. A cab man who drove them to the station told Bartolomei that they had booked to go to Hooton. Bills were issued with their description and a reward of £10 was offered and it seems that William Purcell was wanted for other charges. It is possible he is the William mentioned in a notice in the Liverpool Daily Post 4 July 1870, warning customers not to pay money to a William Purcell, who had been discharged by the Royal Liverpool Friendly Society, although I haven't found anything in newspapers referring to any court case etc.
There is a William Purcell (Parcell) living with his mother and siblings and others at 47 Upper Hope Place, in 1881, who decribes himself as professor of music / musician, but his brother Lawrence is a commission agent:
ED Dist23
Piece 3624
Folio 25
Page number 43
I wondered if she had eloped and if they had fled together to a foreign land. Although there is a Mrs Parcell, the children listed on the passeneger list dd not have a mother's maiden name of cole, so that ruled them out.
William and brother Lawrence in the next censuses seemed to have married - not that is to Elizabeth / Ethelberga.
She didn't seem to be with any of her relatives (parents or siblings), but I wondered I she reverted to using her maiden name. She does not appear to be listed as having died, married or divorced or there being probate, for her under Ethel+ / -other endings under her married name or variations. So she could have reverted to Cole, but a search from 1891 onwards for occupation only as teacher or governess has not brought anything relevant up, other than a possible entry for a widowed Nurse, who I wqas unsure of since I could not find a birth to confirm birth place for your Elizabeth, although she seems to be a twin to Ethelbert who died in 1853 (I think), supposedly born in peckham, but I've only found a registration for Ethelbert.
If her father had been a music professor (at least in the 1851 census), maybe that's how she met William and maybe she would have had connections after her elopement, to be able to find teaching positions, which I guess could have taken her abroad. The other possibility would be if she ended up in some institution (workhouse, asylum or gaol). She could simply have been using a different name and or surname altogether.