Gosh Brandy Snap, what a fascinating story! It sounds like the stuff of TV programmes. I hope your father had a good life despite the obvious sadness.
You've done a lot of excellent detective work and seem to have been very thorough and covered every angle in your research. I can clearly see why you suspect that the deceased son of the wealthy family may be the father - I think I would have come to that same conclusion. It really wouldn't make any logical sense that the family took your 2 year old father into their care only until he completed his education.
I am unable to help with anything specific related to the search, but I have a few thoughts:
There is every likelihood that your grandmother and the son of the wealthy couple met at this lodging house in Cambridge. I take it that the dates tie in? He was lodging there when your grandmother would have visited?
It must have been dreadful for your father to be told not to return home after he had completed his education. I take it he was a boarder?
I note that you have traced family of the deceased young man. If so, I am wondering if a DNA test might reveal something? I am not that au fait with DNA but I believe you will need to find male descendants - so, a male descendant of the brother who you suspect may be your grandfather. I think that some tests may be avialable via the female line too, so you would need to do some research about this. There are many discussions about DNA here on rootschat, but you would have to wade through a lot of irrelevant discussions. As you have a specific connection that you want to find, you could start a new post specifically seeking advice and what test would be suitable for you to confirm this family connection THere are some very knowledgable people here who should be able to advise you. You may like to just cut and paste most of your last post and use it as a basis for starting a new thread. Do you think you may be able to pursuade a family member of the wealthy family to do a DNA test? Are you a male or do you have a brother?
I would look for a will for the deceased young man, though as he was so young he may not have written one.
As the people you are researching are deceased, you can mention their names if you wish.
Regarding your original question - I'm not sure if this is possible, but have you tried looking for records of the Home for Unmarried mothers? Do you know when this home closed down, or when it morphed into something else? Local archives may have records? As your father was not only born there, but also lived there for two years, there should have been some records kept. It is a matter of finding where they are today, if they survive. Maybe Miss Wyrley Birch was your father's godmother because as a member of staff she happened to be present when your father was christened. Are you thinking that she may have known the family who became his benefactors? This is possible too, as it seems that she also came from a wealthy local family, and the families may have know eachother.
A question - was your father's original first name the same as the deceased young man's first name? (I'm a bit confused as you said his names were swapped around). How old was your father when he was christened?
Sorry my reply is a bit of a mish mash, but I just wrote down my thoughts as they came to me. I hope you can follow.
I reckon that DNA would be the best route to go down.

Best of luck with it, and please let us know how you get on.