Sorry, I can't speak to as late as the 1930s, but I have traced relatives who were adopted in the 1920s. I really don't know what the 'closed' adoption system is.
I am a DNA match to a woman born in London in 1950. In helping her to work out how we are related I found that her mum and her aunt were both adopted.
I am related to two sisters (the mum and the aunt of this person), one was born in 1918 and the other in 1919 in London. The mother of these two was a widow who had been born in County Galway in 1888 and was living in London by 1918 after her husband had died in 1913.
By the time of the 1921 Census they were in foster care together a long way away.
In the 1921 Census, each of them were recorded as a 'Foster Child' under their original birth names. When it came to their parents (Column e - orphanhood) it said "NK" - not known. Although I believe that both parents were still alive at this date.
They were adopted at some time after 1921. I have no idea when but I would guess that it was some time in the 1920s. They were adopted by an Irish doctor and his wife who lived in Wembley.
By the time of the 1939 Register, one sister had kept her original forename but the other had it changed. It would seem that the adoptive parents liked one of the names but were not so keen on the other. They both had the surname of the adoptive parents.
One was named Teresa and kept that name. The other was originally named Margaret but by 1939 was going by the name of Joan. At this time the sisters were living above an Italian restuarant in London SW1, one sister was working there as a waitress and the other was a domestic servant.
So, all I can say in response to your question:
"...discovered that the child retained the forenames they were given by the birth parents/family? Or was the adoptive couple very strongly encouraged to change the name given at birth"
I don't know if they were encouraged or not, but one sister kept her birth name and the other sister was given a different name. They were adopted together.