Thanks Colin for bringing this to our attention. I have researched other appointments of the LGC in the London Gazette and they are detailed below. It is quite clear that the Crown's continuing consent to the agreement that allows the rotation of the office-holders is required, and that the Crown, at least in theory, could decline to approve - this may become relevant with the number of small shares, or if one of the number of heirs were no longer a UK citizen / domiciled in the UK.
11 August 1781 [George III, on death of 3rd Duke of ANCASTER AND KESTEVEN]
THE King has been pleased to approve of the Appointment of Sir Peter Burrell, Knight, to exercise the Office of Lord Great Chamberlain of England, as Deputy to the Right Honourable Priscilla Barbara Elizabeth Baroness Willoughby of Eresby, and Lady Georgina Charlotte Bertie, Sisters and Coheirs of his Grace Robert, late Duke of Ancaster and Kesteven, Hereditary Lord Great Chamberlain of England, deceased.
[NB Sir Peter BURRELL was the wife of Priscilla, Baroness Willoughby de Eresby. He was made Baron GWYDYR in 1796, so was Deputy LGC for 15 years without being a member of the House of Lords]
19 June 1821 [George IV, on accession]
The King has been pleased to approve of the appointment of Peter Robert Lord Gwydir to exercise the office of Lord Great Chamberlain of England, as Deputy to the Right Honourable Priscilla Barbara Elizabeth Baroness Willoughby, of Eresby, and Georgiua Charlotte Marchioness of Cholmondeley, coheiresses of the said office
[NB Peter Robert had succeeded his father in 1820 as 2nd Baron GWYDYR and would later succeed his mother as 22nd Baron WILLOUGHBY DE ERESBY in 1828]
1830 [William IV]
Not yet found - should be appointing 2nd Marquess of CHOLMONDELEY.
1837 [Victoria, on accession]
Not yet found - should be (re-)appointing 22nd Baron WILLOUGHBY DE ERESBY
1865 [Victoria, on death of 22nd Baron WILLOUGHBY DE ERESBY]
Not yet found - should be appointing 23rd Baron WILLOUGHBY DE ERESBY
27 January 1871 [Victoria, on death of 23rd Baron WILLOUGHBY DE ERESBY]
THE Queen has been pleased to approve of the appointment of Gilbert Henry, Lord Aveland, to exercise the office of Lord Great Chamberlain of England, as Deputy to Clementina Elizabeth, Dowager Lady Aveland, and Charlotte Augusta Annabella, Dowager Lady Carrington.
[NB Gilbert Henry was 2nd Baron AVELAND, later 25th Baron WILLOUGHBY DE ERESBY from 1888, created 1st Earl of ANCASTER in 1892. He was initially Deputy to his mother (Clementia) and his aunt (Charlotte), until the death of his mother in 1888.]
3 September 1901 [Edward VII, on accession]
THE King has been pleased, by Letters Patent under the Great Seal of the United Kingdom of Great Britain and Ireland, to appoint George Henry Hugh, Marquess of Cholmondeley to the office of Lord Great Chamberlain of England, to hold the said office until such time as His Majesty's pleasure shall be further signified.
Such appointment shall in no way prejudice the claims to the said office which have been referred by the King to the Lords Spiritual and Temporal in Parliament assembled to report thereon, and shall not confer upon the said George Henry Hugh, Marquess of Cholmondeley, any right or title in or to the said office other than he may have had if such appointment had not been made by His Majesty.
24 June 1902 [Edward VII, on report of Committee for Privileges]
The KING has been pleased, by Warrant under His Majesty's Royal Sign Manual, bearing date the 21st instant, to approve of-the nomination and appointment of George Henry Hugh, Marquess of Cholmondeley, to exercise the office of Lord Great Chamberlain of England, such appointment having been made by the claimants to the said office, viz., the said George Henry Hugh, Marquess of Cholmondeley, Gilbert Henry, Earl of Ancaster, and Charles Robert, Earl Carrington, G.C.M.G., in whom, on the finding of the Committee for Privileges of the Lords Spiritual and Temporal in Parliament assembled, the right of selection of a deputy vests.
31 January 1911 [George V, on accession]
The KING has been pleased, by Warrant under His Majesty's Royal Sign Manual, bearing date the 30th instant, to approve of the appointment of Charles Robert, Earl Carrington, K.G., G.C.M.G., to execute the office of Lord Great Chamberlain of England.
[NB Charles Robert, Earl Carrington was created Marquess of LINCOLNSHIRE in 1912]
16 October 1928 [George V, on death of the Marquess of Lincolnshire]
The KING has (been pleased, by Warrant under His Majesty's Royal Sign Manual, bearing date the 18th instant, to approve of the selection of Viscount Lewisham to execute the office of Lord Great Chamberlain of England.
[NB new LGC was not amongst the co-heirs, but instead the husband of one of the daughters of the late Marquess of Lincolnshire. He does not appear to have been appointed Deputy (cf 1781 or 1871). He succeeded as 7th Earl of DARTMOUTH in May 1936, after his appointment as LGC lapsed in January 1936 following the death of George V.]
24 July 1936 [Edward VIII, on accession]
The KING has been pleased, by Warrant under His Majesty's Royal Sign Manual, bearing date the 20th instant, to approve of the Selection of the Most Honourable George Horatio Charles, Marquess of Cholmondeley, to perform and execute the Office of Lord Great Chamberlain of England.
29 January 1937 [George VI, on accession]
The KING has been pleased, by Warrant under His Majesty's Royal Sign Manual, bearing date the 22nd instant, to approve of the selection of the Earl of Ancaster to execute the Office of Lord Great Chamberlain of England
1951 [George VI, on death of 2nd Earl of ANCASTER]
Not yet found - should be appointing 3rd Earl of ANCASTER
13 June 1952 [Elizabeth II, on accession]
The QUEEN has been pleased by Warrant under Her Majesty's Royal Sign Manual, bearing date the 9th instant, to approve the selection of George Horatio Charles, Marquess of Cholmondeley, to perform and execute the Office of Lord Great Chamberlain of England.
3 June 1966 [Elizabeth II, following infirmity of the 5th Marquess]
The QUEEN has been pleased by Warrant under Her Majesty's Royal Sign Manual, bearing date the 27th May, to approve the selection of George Hugh Cholmondeley, Esquire (commonly called Earl of Rocksavage), to perform and execute the office of Lord Great Chamberlain of England.
[NB the Earl of Rocksavage would succeed as 6th Marquess of Cholmondeley in 1968]
14 November 1990 [Elizabeth II, following death of 6th Marquess]
THE QUEEN has been pleased by Warrant under Her Majesty's Royal Sign Manual, bearing date 5th November 1990, to approve the selection of David George Philip, Marquess of Cholmondeley to perform and execute the office of Lord Great Chamberlain of England