Caroline Gurney <research@...>
The Operative Jewish Converts' Institution (OCJI) was founded by the London Society for Promoting Christianity among the Jews (LSPCJ) in 1829. Originally located at Palestine Place, Bethnal Green, the OCJI moved to Bodney Road, Hackney, after Palestine Place was sold in 1895. See: https://missiology.org.uk/pdf/e-books/gidney_w-t/london-society-for-promoting-christianity-among-the-jews.pdf
I suspect that any records which still exist will be in the archives of the Church's Ministry Among Jewish People (CMJ) which is the successor body to the LSPCJ. You can contact them to find out: https://www.cmj.org.uk/contact-us. I'm about to do so myself as, in the course of my research into Bristol's Jewish community, I have discovered two people with links to Bristol who were converted and baptised at Palestine Place in 1835 and went on to become missionaries in Jerusalem and North Africa, ultimately being ordained as Christian ministers.
The baptism records for the Jew's Chapel at Palestine Place can be found in the registers of St Matthew's, Bethnal Green, which are available on Ancestry. The records are faded and poorly indexed, so I had to use the browse function to find the entries I was interested in. It was worth the effort as, for adult converts, the town of birth and names of parents were recorded. In both the cases in which I am interested the men subsequently used their father's given name as a patronymic surname, despite a different surname being recorded in one case (the other had the surname left blank).
In the same way, you should be able to find the baptism of the person you are researching in Ancestry's dataset 'London, England, Church of England Births and Baptisms, 1813-1920': https://www.ancestry.co.uk/search/collections/1558, assuming they were baptised before the end of 1920. I hope the Hackney baptism registers will be as informative as the Bethnal Green ones, in regards to birth town and parentage.