I am in a similar place although not my direct line, it would hopefully lead to my great grandmother's family. I call this "Finding COHENs in a haystack" - so people who went by COHEN in the UK.
I have been researching COHEN (Hyman/Herman and Millie). They started in Vitebsk (pretty sure Sebezh) which was part of Latvia some of the time, and moved to London by the 1890s. They had two children in London, Lewis (1900) and Jack (1903) and all four were in NY by the 1905 census. My grandmother and her siblings listed Millie (as "aunt Musha") as their contact, with a valid address so I found the family from 1905 on. For the most part I have what I can get in the US except for Millie's death record (one possibility).
Herman had a cousin Sol ARISON stay with him. I have more info about him and some of his siblings. He came to the US (Boston>Fall River>NYC>DC) from Sebezh but his father Raphael COHEN (or ARISON or ERSOV/ERSHOV) went to London with a few of his children, some did not want to move. Most of Sol's siblings went by COHEN or some variant so it hasn't been easy to find which Sam COHEN is his brother, but I do have quite a bit about his sister Pearl/Pesla. I have strong DNA matches with a number of Pearl's descendants. Pearl was married to Sam ZICHERMAN (he must be the link to my great grandmother). I have his manifest (as Salmon Sicherman), Pearl's and their oldest son's.
I have searched Jewishgen Belarus, Litvak SIG, Latvia SIG databases with various combinations of names, wildcards, possible COHEN variations like KOGAN and KAHAN. Although I mostly believe that "Aunt Musha" may not have been an actual blood relative, my (actual) aunt referred to Jack as a cousin, so there's that. And Jack spelled his last name as KAHN (New Yorkers may remember his piano business).
So all very confusing and made much more difficult because of the changing surnames. What I can't find that might help:
Birth records for the two sons Lewis and Jack COHEN in the UK. There are a few possibilities for each in the registers but is there any more detail available?
1901 UK Census for Herman/Hyman and Millie/Musha. They were definitely there at the time.
Manifests for any of this family (I might have Herman's or at least there are a few possibilities, but not Millie/Musha or Jack or Lewis) - there is a very narrow window between Jack's birth in Feb, 1903 and the 1905 Census (June) so it must be there somewhere. Likely port was NY or Boston.
ANY records from Latvia/Belarus for the Arison/Ershov/Cohens (specific to this group of individuals) or ZICHERMAN. The ultimate goal for me is to find ZICHERMAN in Vitebsk as my GGM was Feige ZUCKERMAN.
Any records of the ARISON/ERSHOV/COHENs move to the UK - would have been after 1887
Researching Goldner, Singer, Neuman, Braun, Schwartz, Reichfeld (Hungary/Slovakia); Adler, Roth, Ader (Galicia); Soltz/Shultz (Vitebsk, maybe Lithuania)