Thank you Chris, for your help- much appreciated. The time frame I am looking at for this elusive Professor William Bingley is from 1817 to approx. 1843 at the latest. It all ties into another thread I have posted on Elizabeth Playfair, his wife (no marriage record found).
New information shows Eliz. Playfair enrolled at the School for the Indigent Blind in Surrey from 1815 to June 1818. When she left there in June she was 5 months pregnant and the illegitimate child born in October was baptized in November 1818 with William Bingley named as the father. Place of baptism- St. James Westminster. Place of abode- the "Infirmary". While at the School for the Indigent Blind, Elizabeth was noted as having earned a lower wage from her earnings as she had spent considerable time taking music lessons. Our family history notes this Wm. Bingley as a "Professor" (I strongly doubt that he was a "real" Professor), thus my search for a musician or teacher of music. Possibly, Bingley had nothing to do with music, at all.
The poor law/workhouse for St. James parish was on Poland Street and according to the online catalogue for the City of Westminster Archives, 1818 admissions/discharges of inmates are missing. But perhaps still worth a call or email to them to discuss other records.
The Rev. Wm. Bingley was a noted author and cleric at St. Georges Bloomsbury, London but in 1818 he was married with a young son. Although he could "loosely" be referred to as a "Professor" I doubt that he would have fathered an illegitimate child under his circumstances. It probably would have been a major scandal of the day, in 1818.
