Ok, now I understand re CMJ. London Society for Promoting Christianity among Jews was particularly active in the 1850s/60s in London, trying to convert Jews, especially those who were struggling making a living. What are trying to establish in their papers? If conversion, you might try parish records for the church where Raphael Marks lived.
Marriage certificate. There were 2 separate documents regarding the marriage - one a governmental record, the other - ecclesiastical. If he was converted, the marriage would have taken place either in his parish or in the parish of his bride. That might throw some clues. The parish records are normally kept in the local records office. If lucky, they will be digitised, and you can search them online.
Another point about CMJ is that many Jews saw conversion as means of obtaining free education and/or improve their conditions. They might have still practised Judaism. Jewish Chronicle had some articles and letters about this around that time.
Re the reasons of moving from Russia (most likely Russian Empire, which would have included Poland, Ukraine and Lithuania) could be several:
a) Crimean war - conscription, POWs, etc.;
b) desire to avoid conscription into the Russian Army (if conscripted, men served 25 years);
c) antisemitism in the Russian Empire. There was a steady stream of immigration in the 1840s/50s from Russia to England, with London as the biggest attraction, because of the large and established Jewish community and the opportunities, real and perceived, to make a living.
I would suggest to contact:
a) local record office (local to the place of his residence). In addition to parish records, there might be some others, which might provide a clue to his place of origin.
b) National Archives - they'll have immigration records; naturalisation records; incoming passengers.
Also, check Ancestry database and FindMyPast database. I find that combining these databases works best.
Check all census returns (earliest 1841, which provides minimum information, but later ones give you much more.) You might find that the column "where born" might differ, and might narrow down the search (For example, 1851 census might state "Russia", but 1861 - "Poland".)
Also, they have immigration and incoming passengers records.
Hope this gives you some leads, and apologies, if I duplicate the search you've already done.