In his answer, Peter Shor has set out the wording in the text of Adonais that suggests the identification of the mourners.
But we don’t have to rely only on the text: there is also the epitext, the “textual materials that derive from a published work, e.g. reviews, interviews, etc., that contextualize the work and shape its public reception” (OED). In the case of Adonais there is a tradition of interpretation that we can look at: in particular, as soon as the poem was published, reviewers were confident in identifying Byron and Moore:
In the course of the poem some living writers are introduced, among whom Lord Byron is designated as
The Pilgrim of Eternity, whose fame
Over his living head like Heaven is bent
An early but enduring monument!
The poet of Ireland† is called, with equal brevity: and felicity,
The sweetest lyrist of her saddest wrong
Anon (1822). ‘On Mr Shelley’s New Poem, Entitled Adonais’. In The Examiner, number 757, 7th July 1822, p. 420.
† See below for evidence that “poet of Ireland” refers to Moore.
This shows that Shelley’s hints in Adonais were sufficiently clear for his contemporary readers, even if these hints seem obscure at a distance of two centuries.
Shelley himself confirmed that Byron is mentioned in the text, though he did not say explicitly where:
Lord Byron—I suppose from modesty on account of his being mentioned in it—did not say a word of “Adonais”, though he was loud in his praise of “Prometheus Unbound”: and, what you will not agree with him in, censure of “The Cenci.”
Percy Bysshe Shelley (26th August 1821). Letter to Leigh Hunt. In Roger Ingpen, ed. (1914). Letters of Percy Bysshe Shelley, volume II, p. 910. London: G. Bell.
Other writers took up “pilgrim of eternity” as a sobriquet for Byron, indicating that they too understood Shelley to be referring to him in Adonais:
Byron, had I met him, instead of sending Finlay, would have been at Salona now. His name was the means chiefly of raising the loan in England. Thousands of people were flocking here: some had arrived as far as Corfu, and hearing of his death, confessed they came out to devote their fortunes, not to the Greeks or interest in the cause, but to the noble poet; and the pilgrim of eternity having departed, they turned back.
Edward Trelawny (29th April 1824). Letter to Leicester Stanhope. In Leicester Stanhope (1824). Greece in 1823 and 1824: Being a Series of Letters and Other Documents on the Greek Revolution, p. 331. London: Sherwood, Jones and Co.
In our days it [Corinth] has still been a scene of contest: “Corinth is taken, and the Greeks have gained a battle in the Archipelago,” says Lord Byron, in a letter to Mr. Kinnaird, written in Greece a few months only before the warrior poet had become the “Pilgrim of Eternity.”
William Brockedon (1833). Finden's Illustrations of the Life and Works of Lord Byron, volume I. London: John Murray.
How can I be confident that by “poet of Ireland” the anonymous reviewer in The Examiner, quoted above, meant “Thomas Moore”? Well, this phrase was a common sobriquet for Moore, meaning that he put the national sentiments of Ireland into verse. There was even a minor critical controversy over whether he deserved this name.
This piece [‘The Tear’], which were we to call merely beautiful, it would reflect little credit on our poetical taste, is from the pen of Thomas Moore, the celebrated living poet of Ireland.
Peter Forbes (1816). The Pocket Encyclopedia of Scottish, English, and Irish Songs, volume II, p. 37. Glasgow: Andrew & James Duncan.
Like you I admire the lively and graceful genius of this man; like you I appreciate the amiable temperament and dispositions which lend a charm to his verses, more touching than any thing which liveliness, grace, and genius alone could confer; but I cannot consent for a moment to class Mr Moore with the great poets of England—no more can I persuade myself that he is likely to go down to posterity as the national poet of Ireland. The claim which has lately been set up for him† is one of no trifling import, it would not only assign to him a share of the same magnificent honours which have of right descended to Byron, Wordsworth, and Campbell, but mingle with his laurels another wreath such as the grateful affection of your own country has already woven for Scott and Burns.
‘Baron von Lauerwinkel’‡ (1818). ‘Remarks on the Poetry of Thomas Moore’. In Blackwood’s Edinburgh Magazine, October 1818, p. 1.
† That is, although the critic objects to the sobriquet, he admits that it is widely applied to Moore. ‡ A pseudonym of John Gibson Lockhart.
He [Pierre-Jean de Béranger] has been compared to Moore, which is doing much injustice to the national poet of Ireland.
Anon (1823). ‘French Poets of the Present Day’. In Blackwood’s Edinburgh Magazine, May 1823, p. 514.