In terms of both second name changes, Baum is a common name in Manchester - one of my Abrahams/Abrams relations married a female Baum or Boam (interchangeable) in Manchester in the late 19th century.
I have heard of Berens becoming Brown, Brand or similar.
In terms of first name changes, in my family Harris or Hirsch often became Harry. Aaron also became Harry.
Commonly families reverted from their surnames to their patronyms. My great great grandfather was Mordecai/Mortchel, and this was the family patronym for my father's family so some members of the family changed their surname from Servian/Serwianski to Max, Marks or Maxwell after Mordecai . This was in both UK in Liverpool and in USA in Chicago.
My Abrahams/Abrams family was called Ceglarski in the old country but reverted back to the patronym of Abram/Abraham in Manchester.
But occupations could also be used. One Servian became Silverman as he silvered mirrors, but he also later became a Maxwell.
Another Servian borrowed a neighbour's name, unrelated to the family and became Lynam.
So you need to think outside the box and examine census and BMD records for clues. It took me ten years to find out some of the above, but the clues were there in the records.
Jill Whitehead, Surrey, UK