I agree with the attribution to Henry Walton. There are several characteristics that I find typical of Walton. The immediacy, the direct gaze, the handling of the paint.
This face is particularly like the face in the portrait of Henry Walton's sister, Elizabeth Bridgman. An enlargement of this portrait is available on Wikimedia Commons.
Alleyne's paintings, on the other hand, seem to have a gentler quality and are a little unsure in the anatomy.
Are there any comparable examples of Alleyne's work?

Many of Alleyne's portraits depict his sitters seated. Furthermore, Alleyne often includes part of the seat furniture in his portraits. I don't believe Walton generally does that. These examples all strike me as very close to the Holt picture in conception and style

Of the two attributions suggested, Francis Alleyne or Henry Walton, I think that Alleyne is more likely for the reasons given by Miles. The portrait is very different to Walton as I know him and I don't think the comparisons above stand up to scrutiny. Walton's colouring, his formats and his face structures are unlike our portrait.

I’ve attached a composite based on another portrait of Rowland Holt on Art UK (by John Downman, at Gunby Hall). Holt certainly looks much older in the Downman portrait than in the work that is being discussed.

https://artuk.org/discover/artworks/rowland-holt-of-redgrave-suffolk-171424

Here is another link to the portrait by Downman.

https://www.nationaltrustcollections.org.uk/object/637625

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The attached snippet from 'British and Irish Paintings in Public Collections' (2006) by Christopher Wright and Catherine May Gordon shows that the title of this work used to be 'Rowland Holt of Redgrave Hall'.

The current title could be improved.

***

Here are portraits of two members of the Holt family of Redgrave – likely two of Rowland Holt’s siblings:

https://tinyurl.com/z78t2rxb
https://tinyurl.com/mr29j5p3

Wikipedia shows that Rowland Holt's dates were 1723-1786.

https://tinyurl.com/5c93mxsd

His burial record on Ancestry also includes those dates.

An extract from an article in the 'Newmarket Journal' of the 21st of September 1935 mentions the acquisition of this painting. It was donated by the three sisters of Prince Frederick Duleep Singh (1868-1926).

https://tinyurl.com/25y83tns

Could the artist be John Trumbell (1756–1843)? He was an American artist who worked in London and who studied under Benjamin West.

Please take a look at this painting at the White House 'Thomas Jefferson' (1788). The text indicates that Trumbell had first painted the image as part of another work and he copied it for this one.

https://www.whitehousehistory.org/photos/thomas-jefferson-by-john-trumbull

See also, https://smarthistory.org/trumbull-declaration/

I'll look in the NPG archive next week if I can. I think we are moving towards closure with an attribution to Francis Alleyne. But I'd like to strengthen the evidence.

Alleyne's life and work are so little known that it is worth summarising Ellis Waterhouse's summary biography in his Dictionary of British 18th century painters: Active as a portrait painter between 1774 and 1790, usually on a small scale. Exhibited a single portrait at the Royal Academy and the Free Society of Artists in 1774 and at the Society of Artists in 1790. Seems to have specialized in small oval ¾ lengths usually signed on the back. Probably moved around Kent in 1786, going from family to family to paint portraits.

Our portrait is indeed oval and is painted at Alleyne's favoured size of about 36 or 37 by 29 or 30 cms.

To respond to the discussion question, "Is this portrait of Rowland Holt by Francis Alleyne or Henry Walton?" It is clear that it is not by Walton, as is evident when looking at Walton portraits on ArtUK where our portrait is the odd man out, and also for the reasons that I set out in an earlier post (12/01/2022).

Miles Cato (26/09/2018) identified that many of Alleyne's portraits depict his sitters seated, often including part of the seat furniture in his portraits. I noted that our portrait is oval and is painted at Alleyne's favoured size of about 36 or 37 by 29 or 30 cms. None of this is not conclusive in itself.

I had a look in the NPG archive at the file of portraits by or attributed to Alleyne. The pose and composition match several autograph works by Alleyne. The face is craggier as a depiction of an older man than most of his sitters. In my opinion an attribution to Alleyne is appropriate.

I’d be interested to know where ArtUK sourced Alleyne’s life dates as 1750-1815.

Jacob, those are the dates given for him in Benezit's dictionary. He does not have an ODNB entry. The NPG only gives his dates of activity.

The life dates 1750-1815 are not in the 2006 English edition of Benezit. So where are they?

Here are the records from Ancestry for a Francis Alleyn who was baptized in Richmond in 1739 and who passed away there in 1815. The baptism record has a link to a painting site so I thought this man might be the artist Francis Alleyn.

Interesting. The signature in the last Alleyn marriage attachment above matches that in a 1774 marriage, attached below, but in 1774 with the addition of a final 'e' making it 'Alleyne' - by now a widower. The particular interest of the 1774 marriage is Alleyne's second wife, Elizabeth Roth, who could be the daughter of Roth the drapery painter, not a common name. I could dig a bit further on Roth, perhaps.

What we have, thanks to Marcie, is a birth and death record for one man and marriage records for another. Or are they one and the same? Evidence?

Jacob, if you go to Oxford Art Online and look for Alleyne, you will see the following:

Alleyne, Francis
Published online: 31 October 2011
Collection: Benezit Dictionary of Artists

Alleyne, Francis British , 18th century , male. Born 1750; died 1815. Painter. Portraits. Francis Alleyne was active from 1774 to 1790. He exhibited at the Royal Academy and the Free Society of Artists in 1774 , and at the Society of Artists in 1790. He specialized in oval, three-quarter length portraits...

And the source for these life dates, added to Benezit between 2006 and 2011? That is the next question.