Hi Allan
In the absence of a father's name on your original birth certificate and the fact your birth mother has passed away taking this information with her, there is no way of establishing the identity of your putative father within the records currently available to you. If you have seen the contents of your adoption file and that information is not there, no professional researcher or any amount of money will be able to help or solve your query.
Family historians use a variety of sources in addition to the statutory birth, marriage and death indexes and the censuses. In your case I would be looking for telephone directories and street directories such as Kellys as you have suggested a location and occupation for the man you seek. It might also be worth your while checking the parish registers for a baptism prior to your adoption. Some parish priests were inclined to name the putative fathers of illegitimate children.
Unfortunately you have not updated your profile so we don't know where in the world you are and if you are able to travel. So suggesting a day trip to The National Archives at Kew where they hold a vast collection of directories in the library on open access, or the BT Archives at Holborn for phone books, or the Centre for Buckinghamshire Archives at Aylesbury for other sources may be a waste of time.
Electoral registers weren't compiled for the duration of WW2 so again another source not available to you. It is hoped that the 1939 National Registration Index goes online later this year with FindMyPast but I am led to believe that will be name driven, information you don't have. The next census due for public release will be the 1921 census in 2021.
As a final resort, you could take a DNA test but they can be costly and I note you have limited funds and the results may not answer your questions.
I realise that this search is important to you but with limited information there is little we can do to help further at this stage.
Dawn
