Alexander Schmidt, Shakespeare Lexicon and Quotation Dictionary, third edition (1902), observes that Shakespeare uses cormorant four times in his plays—once (in Richard II) as a pure noun and three times (in Love's Labour's Lost, in Troilus and Cressida, and in Coriolanus) adjectivally. Here is Schmidt's entry for the word:
Cormorant, a glutton: light vanity, insatiate cormorant, consuming means, soon preys upon itself, R2 II, 1, 38. Adjectively, = ravenous: cormorant devouring time, LLL I, 1, 4. in hot digestion of this cormorant war, Tril. II, 2, 6. the cormorant belly, Cor. I, 1, 125.
In effect, Schmidt interpolates a comma after cormorant in the line from Love's Labour's Lost to yield a sense along the lines of "ravenous, devouring time."
Most early Shakespeare annotators don't offer a gloss on "cormorant devouring time," but it's interesting that a writer in The Lancet (January 1854) provides the missing comma in his invocation of the phrase. From William Harding, "On Tetanic Spasm and Its Treatment by Chloroform":
... moreover it [tetanus] occurs for the most part in the prime of life, in the vigour of healthful manhood—to those who at the time of its abrupt invasion were enjoying healthful ease, and before the elasticity of health has been enfeebled or destroyed by "cormorant, devouring Time."
The same comma pops up less than four years earlier in George Throop, Life-Leaves: From a Rover's Log, serialized in Sartain's Magazine (October 1851):
Hardly conscious of hat I did, I went nearly to the end of a neighbouring wharf, both sides of which were thronged with vessels. One of these, a dilapidated old thing, on which
"Cormorant, devouring Time"
had apparently done his worst, was getting under way. Her mainsail was already up, and a man was casting off the stern-fasts.
"The Literary Garden" in The European Magazine, and London Review (September 1815), in contrast interpolates a hyphen:
Where, spite of cormorant-devouring Time,/ The' endeavor of this present breath, may buy, That honour, which shall bate his scythe's keen edge,/ And make us heirs to all eternity.
But singling out cormorants to be devoured by time seems inexplicable, whereas using them as a figure for ravenous hunger will make sense to anyone who has watched a cormorant come to the surface of lake or inlet, holding a small fish in its beak, and then swallow the victim whole.
Hyder Rollins, A New Variorum Edition of Shakespeare: The Sonnets includes a footnote in which the editor notes that time as a ravener was a longtime theme in Western literature:
VERITY (ed. 1890) thinks the line [line 1 of Sonnet 19—"Deuouring time blunt thou the Lyons pawes"] may be a reminiscence of Ovid, Metamorphoses, XV.234, "tempus edax rerum, tuque, invidiosa vetustas." He also cites Spenser's Amoretti, 1595, sonnet 58 (1908 ed., p. 728), "Devouring tyme and changeful chance have prayd Her glories pride."—LEE (ed. 1907): Another echo of Ovid's philosophic argument ... {"tempus edax rerum," etc.} which Golding translates {1567, XV.258 f. (1904 ed., p. 300)}: "Thou tyme, the eater up of things, and age of spyghtfull teene, Destroy all things."—ALDEN (ed. 1916) compares Daniel, Delia, 1592, sonnet 46 (1930 ed., p. 33), "times consuming rage."—TUCKER (ed. 1924) compares Love's Labour's Lost, I.i.4, "cormorant devouring Time."—TRAVERSI (Approach to Sh., 1938, p. 46): The sense of the hostility of Time is fundamental, not only in the Sonnets, but to the plays of this period. ... The theme indeed, was a commonplace of the age; it was associated with the Platonizing philosophy adopted by the court poets, and with the religious 'pessimism' of mediaeval tradition.
It seems clear to me from these various sources that "cormorant devouring time" refers not to a special period during which it is appropriate to devour cormorants, nor to a special predilection on time's part to consume cormorants, nor yet to a cormorant's otherwise unrecognized ability to eat time, but to time as both ravenous and devouring, with cormorant standing in for ravenous as an adjective because cormorants themselves were seen (in Shakespeare's day) to be voracious eaters of their finny prey.