The time spent watching @arthistorynews may have paid off.
I think I have a very strong candidate for "Mr Holland" of Wimpole Street painted by George Romney in 1785/6
@EJBrand @isherwood_FA

London Tax Records for Wimpole St are frustratingly unavailable for 1785-9. But I found "William Holland, esq" at 28 Wimpole Street 1790-4

In 1784 Sir William Pepperell is at this address and in 1795 Gilbert Franklin, esq is listed. Where is William?

He appears again in 1799 at "17 Great Cumberland Place" and he appears here regularly until 1812 although sometimes under the variant spelling "Hollond"

A search of "Wm Holland" + "Great Cumberland Place" brings up the June 1803 edition of "The Gentleman's Magazine" where we read in births "In Great Cumberland Place, the wife of Wm Holland, esq, a son."

Searching for a male Holland born in this area june 1803 gives us the baptism of "Frederick Hollond" 5/7/1803 born 9/6/1803. Parents are William and Harriet.

A search of marriages finds they were married 13/9/1796 at St George Hanover Square. Harriet's maiden name is "Pope".

These names were the key: Searching their names together brought up family trees and some other information but best of all @DeborahACohen book 'Family Secrets: Shame and Privacy in Modern Britain.

Here we learn William was a writer in the East India Company. He comes to England in 1784 with the huge sum of £100,000 and two "Eurasian sons, Henry and Charles". He is 34 and looking for a wife.

By 1786 it looks like he has found the perfect woman, but only on the condition he renounces his Eurasian sons.
"Holland refused, his "pride revolting at shutting my doors against two boys whom I have acknowledged & allowed to come home"."

The 19 year old Harriet Pope becomes his lover and they have several illegitimate children before quietly marrying in 1796 legitimising their final three sons who go on to become pillars of the establishment.

(There is so much more to the story, but you really should read it in @DeborahACohen book. It's available as part of the preview on Google books)

So, if I'm correct, William arrives in London in August 1784 (hence not in earlier tax records). He has his potrait painted by Romney soon after he moves into Wimpole Street as he begins his search for a wife.

I searched for other potraits of William but came up empty. However his son Robert, an early hot air balloon pioneer, is in two in the National Potrait Gallery. One by John Hollins...

Is it just me...?

The dates seem to fit perfectly and the story does too. His age (34/35) looks right...it all comes down to whether the Wimpole/Great Cumberland Will esqs are the same person.

I am pretty sure they are, but I can't be 100%.