The London 1862 tournament book has <S. Solomons>, and The Chess Player's Chronicle 1856, 1861, 1862 and 1882 has <Mr. Solomons>.

Hilbert, Writings in Chess History, page 93
Sergeant, Century of British Chess, page 94
[CPC], vol. 4, May 1856, page 140
Spinrad, Jeremy P. [Collected results, 1836-1863]

The two latter has <Solomons>. Hilbert and Sergeant I don't know.

Then comes New York 1857, where <Moses Solomons> is well established.

Then again London June-Oct. 1862, where Edo Chess refers to

Di Felice, Chess Results, 1747-1900, page 36
Gaige, Chess Tournament Crosstables I, page 4
Sergeant, Century of British Chess, page 116
[ILN], vol. 40, 21 June 1862, page 644
[ILN], vol. 41, 19 July 1862, page 91
[ILN], vol. 41, 2 Aug. 1862, page 119
[ILN], vol. 41, 30 Aug. 1862, page 251
[ILN], vol. 41, 11 Oct. 1862, page 395
Spinrad, Jeremy P. [Collected results, 1836-1863]
Feenstra Kuiper, 100 Jahre Schachzweikämpfe, page 106

Of these, Gaige cites the tournament book which has <S. Solomons>. Spinrad has <Solomons>. The rest I don't know.

Is there any proof that the London player is Moses Solomons?

From passenger lists:

<Moses Solomons>, 21, England, upholsterer (?), arriving in New York 14 July 1856. That is right after the London 1856 tournament. I. e. born ca. 1835.

<Moses Solomon>, 28, England, Broker, arriving in New York 9 Sep. 1862. That is right after the London 1862 tournament. With him is Rosetta Solomon, 25 and three children, Catherine 4, Sarah 3, and Emanuel inf. I. e. born ca. 1834.