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A good source for the history of this statue is Sir William Treloar's 'A Lord Mayor's Diary', 1906 - 07,  published in 1920. Discussing the Mansion House he wrote: "The site upon which it stands was originally one of the markets of London, called the Stocks Market, which took its name from a pair of stocks erected there for the punishment of offenders in the year 1281. It was here, in the open place, that Robert Vyner, Lord Mayor, erected in 1675 an equestrian statue of  Charles II, trampling on Oliver Cromwell at all  events, that is what Vyner called it; but when he  bought it, it represented John Sobieski, King of Poland, trampling on a Turk. It was altered and  put up by this loyal Lord Mayor, who was afterwards  made a baronet by a grateful Sovereign. This statue was taken down to make room for the Mansion House to be built in 1738. It remained in an inn yard in Aldersgate Street until 1779, when, by  a resolution of the Court of Common Council, it was given to Mr. Robert Vyner, a descendant of the clever Lord Mayor, and it is now standing in the park at Newby Hall, Ripon, where I have seen it.  I was allowed to take a photograph of it, which I  have reproduced."

English 18th Century Portrait Sculpture adds the following: "On 29 May 1672 the most conspicuous outdoor statue of the time was unveiled by the new water conduit at the Stocks Market near Lombard Street. ... of White Genova Marble, and bigger than the Life ...carved in Italy by an unidentified sculptor. It was shipped to England and acquired by a London merchant, Sir Thomas Vyner, whose nephew, Robert, probably conceived the idea of transforming it into a statue of Charles II trampling on Oliver Cromwell. Latham carved the new head." "The statue was made in Italy .... It commemorated his {Sobieski's} victory in Vienna. The statue was bought in 1672 by Sir Robert Vyner ... who brought it to the Stocks Market, London at the Restoration. Sir Robert had the head refashioned by Jaspar Latham - .... to represent Charles II and the lower figure represented Oliver Cromwell.  ... the statue was removed to an inn yard, then to the Vyner estate in Lincolnshire. Lady Mary Robinson of Newby married Henry Vyner and inherited Newby Hall in 1859. The statue was brought to Newby Park in 1883."

In Lincolnshire the statue was at Gautby Hall, the ancient seat of the Vyner family (destroyed in 1874). There it resided on an island in a lake until being moved to Newby.

Our Wikipedia image shows the statue at Newby Park. The close-up image comes from English 18th Century Portrait Sculpture which has many other images including etchings of the statue in its original location.