These photographic portraits of Southend mayors of the 1920s and 1930s might come in useful another day.
https://bit.ly/2Brzk89

Are there portraits, painted or photographic, of the mayors for those years?

Judging by the hairstyle of the sitter (the centre parting) I think a date of early to mid 1950s is most likely.

Personally I would hope they still have a reference library, with local history section and I would email request and photo to the chief/senior librarian. Failing that 1) write to local paper. He has a distinctive face and someone will remember him. Failing that write to the office of the current incumbent

Perfectly good suggestions - but why don't you do it, Roberta? That's how it works here: apart from Marion who runs things, all of us here are unpaid, many are wholly unqualified, and we do the research we do because we think it needs doing...and because it's a fascinating challenge! Why not join us?

List of mayors of Southend:

https://www.southendmayorscharity.net/wp-content/uploads/2018/05/Mayors-Since-Date-of-Incorporation-1.pdf

Narrows down the field.

Thanks, Maria, but the link to that list is in Marion's introduction at the top.

The mayors of Southend used to provide a forword for the yearly Carnival Programme, plus their photo. https://bit.ly/2Bb04K0

Following Grant's suggestions to focus on the mid 1950s, the attachment shows images of the mayors between 1950 and 1965.
Two are missing: Herbert Henry Smith (1956-57) and Frederick Harry Woods (1957-58). For these years, no carnival programmes are available.
In my opinion, none of those listed in the pdf looks like our sitter and I can't find images of the missing two in other sources (BNA, general google search).

And here is the attachment.

I contacted the mace bearer Adam Tregoning, with our image and after some consideration, he sent the attached photograph of Albert Victor Mussett, JP, mayor of Southend 1959-60.

The photographic image supplied by Adam Tregoning is very much appreciated. The image of Mayor Mussett shows a man also with a centre hair parting, presumably in the late 1950s, which confirms I think that the sitter for 'our' sculpture is from that period. However, I am not sure that there is a good likeness between to the two men as the bone structure appears to be quite different.

I notice that all the Holman works held by Southend were by gift from the artist's sister, Ivy Eliza Holman (1906-1995). Presumably Miss Holman left no notes in regard to this particular piece either by will or otherwise?

I have checked the records of the Royal Academy but this sculpture was not exhibited there. George Holman was a regular exhibitor at the Royal Society of British Sculptors and was elected a full member (FRBS) in 1955. As it is quite possible that he may have exhibited a commission of this kind at the RBS (as a Not For Sale item) I will check the RBS catalogues I have for that period but I do not have anything like a complete set. If others have those catalogues for say 1950 to 1975 it would be very helpful if checks could be made.

According to the ‘Mapping the Practice and Profession of Sculpture in Britain and Ireland’ website, it mentions that George Alfred Holman was active between the years of 1942 - 1948.
Although the 1939 register also records him as being a Sculptor and jeweller.

https://sculpture.gla.ac.uk/view/person.php?id=msib4_1273507376&search=Holman

If this is taken at face value, then does that mean the timeline, due to the intervention of the war, only covers two or a maximum of three sitting Mayors (with a cusp).

1939/45 - William Miles
1945/49 - Stephen Frost Johnson JP
1949/50 - Sidney Henry James Bates

William Miles also has a portrait on ArtUK
https://artuk.org/discover/artworks/alderman-william-miles-cbe-jp-mayor-of-southend-on-sea-19221924-19361937-19391945-2992

Stephen Frost Johnson JP is a bespectacled gentleman and Sidney Henry James Bates is slightly out of the time period, but maybe worth a look.

Is there a similiarity to William Miles?

Thanks for the information above and the comments. I think it quite probable that this piece is a relatively early work but in my view not necessarily restricted to the years up to 1950. George Holman died in 1980 and he was working actively as a sculptor in the 1950s and 1960s and probably later than that.

This work gives every appearance of being a commission, either from the sitter or by the council, yet it remained in the possession of the artist, and then his sister, until it was given to Southend Museums. It begs the question was it rejected for whatever reason or was the artist a friend of the subject and asked him to sit for him? We shall probably never know!

I do not think our bust is of Mussett or Miles; they both look rather less hearty or more delicate. It may be that the bust was rejected by the council as being too quirky or informal for an official portrait.

Is it known in what year Ivy Eliza Holman gifted the bust to the Southend Museums Service?

The way to solve this one is for Southend Museums Service to publicise an image locally (local newspaper/media etc). Only local memory -family or otherwise - is likely to recall who this is, and perhaps its circumstances if controversial, which it might have been. Its a curious mix of technically good modelling, odd proportion and a rather offputting facial appearance: what was probably intended as a good-humoured smile has come out more like self-satisfied smirk. The weird bust of the footballer Cristiano Renaldo that briefly appeared in the airport on his native Madeira comes to mind as a more recent parallel.

Kieran - Ivy Holman also donated three medals and 12 drawings for medals and coins by her brother to the British Museum in 1993 (eg):

https://www.britishmuseum.org/collection/object/P_1993-1002-52

So it's likely the donation to the Beecroft would have been then or before that. The Beecroft has 21 pieces by Holman - mostly plasters, so Ivy might well have gifted these before 1993 (though after her brother's death in 1980) due to the number and size of them. But it looks like she wanted to preserve her brother's legacy before her own death in 1995.