Hi, it seems you have quite a problem with the Oeschgers, Osgars, Oscars, Paynes, etc.
In the 1841 Census there is a May Osgar (1790) with daughter Caroline (1822) both born in Middlesex.
But did you notice that in the 1891 Census, Caroline (mother of Head) gives her birth as Middlesex? Here the name is Oeschger which is not only a Swiss name but a German, too.
However, the place of birth is not important for searching ancestors in Switzerland. The records are kept in the town where the person's ancestors originally came from.
To give you an idea: My husband was born in St. Gallen in the Eastern part of Switzerland, as was his father, BUT the family originated from Galgenen in the Kanton (County) Schwyz and here are all the family records. The problem here is that the records are written in the old script (Sutterlin or other old scripts) which one has to learn to be able to understand the them.
It's not impossible and I think more online data are being made available and there is a wonderful Old Scripts Tutorium developed by the University of Zürich for those who can speak and read German:
http://www.adfontes.uzh.ch/1000.phpwhich gives online courses in reading the various Old German scripts (not just for Switzerland).
So you see, even if people would like to help you, without more information, it will be difficult especially as the name changes and the birth of Caroline Oeschger is maybe not Switzerland.
Maybe someone has an idea, though.....