Southwark and Westminster registers are on Find My Past.
No register from Dorset or Hampshire in county list in those dioceses on FindMyPast.
It says on Southwark Diocese website (my previous reply) that parish registers until 1910 have been digitised and are on FindMyPast.
How do you know that Elizabeth was baptised at the Oratory in Bournemouth?
History of the Oratory Church, Bournemouth, Church of the Sacred Heart
https://www.bournemouthoratory.org.uk/church-historyA wooden chapel was opened in 1870 on the site of the present Sacred Heart. Baptism registers started 1872. Parish priest was finding it difficult to cope on his own with the growing population in 1870's.
When there had been only a few Catholics in Bournemouth and no permanent chapel, they travelled to St. Mary, Poole.
Sacred Heart is mother church to 7 churches.
Elizabeth may have put Bournemouth as her birthplace after she moved away because it was the nearest big town that a person on London or Manchester might have heard of. Like when I'm trying to explain to people where I'm from, I sometimes have to say "north of Manchester".
Other Catholic churches near Bournemouth.
https://www.genuki.org.uk/big/eng/HAM/Holdenhurst/churchSelect "Nearby Churches" tab and select a distance for a map which shows locations of churches within a chosen radius. Catholic churches have yellow markers. Click on a marker for name of church. Unfortunately foundation year isn't shown on name tabs. You would have to google each one to find out which existed in 1870s before you can search for location of registers.
