The Red Admiral (Vanessa atalanta) and White Admiral (Limenitis camilla) butterfly species cannot have had those common names before 1627, when the English Navy (the predecessor of the Royal Navy) was first divided into red, white and blue squadrons and admirals began to be referred to by the colour of their squadron. Those butterflies could be found in England before that date, so what were they called?
Kaspar von Greyerz, Connecting Science and Knowledge: Scenes of Research (2013) has an interesting comment on the subject of common species names in the seventeenth century:
While it is beyond my scope to give a thorough analysis of folk species in early modern languages, the evidence suggests that late medieval and Renaissance Europeans were generally content to refer to insects with the name of their folk genus: dragonfly, butterfly, beetle, and the like; only in particular instances, such as the ladybird or cockchafer, did they descend to the taxonomic level of folk species. Aldrovandi, for instance, enumerated 117 different butterflies and moths, some of them males and females of the same dimorphic species, but most of them separate species. But he did not name them. It is likely that they did not yet have vernacular names. The English common name "Admiral," for a particularly striking butterfly, is attested first in 1699. In his Monatlich herausgebebene Insecten-Belustigung, published in 1740 to the late 1750s, August Johann Rosel von Rosenhof gave vernacular names to butterflies and other insects that were simply descriptions, beginning his work with "the big social thorn caterpillar with gold-red spots." Though the research remains to be done, I suspect that most vernacular names for folk insect species were developed by collectors, beginning in the seventeenth century, in parallel with the development of scientific taxonomy.
Peter Marran & Richard Mabey, Bugs Britannica (2010) [combined snippets] offers the following background on the name admiral in connection with the Red Admiral butterfly:
There has been much debate about this butterfly's name, which is widely held to be a corruption of admirable: hence, the 'Red Admirable'. That view was shown to be wrong by Maitland Emmet, who found literary references to 'Admiral' that long preceded the use of 'Admirable' and proposed a convincing explanation of what Admiral actually meant. In one of his early eighteenth-century insect 'catalogues', James Petiver defined admirals as 'such butterflies as generally have a white, yellow or other field in the midst of their upper wings; and the rest for other colours' (by 'field' he meant a patch). The obvious analogy is with a naval flag. When an admiral was on board his flagship, an ensign was hoisted consisting of a plain field with colours in the corner. The colours of this butterfly were reminiscent of the Union Jack, the flag of the United Kingdom, which had come into being around the time Petiver was cataloging his butterflies. Admiral may, however, be a much older folk-name, for we share it with Germany, Sweden and the Netherlands.
Admirable, by contrast, first surfaces in 1749, when Benjamin Wilkes, a semi-literate artist and engraver, guessed that this might be the butterfly's original name, from its 'great Variety and Beauty of its colours'. Subsequent authors jumped to the conclusion that Admiral was only a corruption of Admirable, but they had the facts in reverse. In the Channel Islands, the butterfly is still known as the Alderman, perhaps another corruption of Admiral, while in Cornwall, Essex and the Black Country, it was sometimes called 'King Georgie' (along with the Peacock and Small Tortoiseshell).
With regard to the word admiral itself, Merriam-Webster's Eleventh Collegiate Dictionary (2003) reports that the word has been in English since the fifteenth century, with the original (now archaic) meaning "the commander in chief of a navy."
Michael Salmon, The Aurelian Legacy: British Butterflies and Their Collectors (2000) includes an appendix titled "List of British and Irish butterflies together with their past and present common names." This list includes the following coverage of the two admirals:
THE WHITE ADMIRAL (Ladoga camilla (Linnaeus, 1764)): Petiver, 1703.
The White Leghorn Admiral (Petiver 1703)
The Leghorn white Admiral (Petiver, 1706)
The white or White Admiral (Petiver, 1717; Berkenhout, 1769; Haworth, 1803; and some subsequent authors)
The White Admirable (Wilkes 1741–42; Harris, 1766; Brown, 1843; Newman & Leeds (1) 1913)
THE RED ADMIRAL (Vanessa atalanta) (Linnaeus, 1758): Moffet, 1634 (Fig. 151); Merritt, 1666.
The Admiral (Petiver, 1699; Ray, 1710; Albin, 1720; Dutfield (1), 1748; Berkenhout, 1779; Haworth, 1803)
The Admirable (Wilkes, 1747–49; Harris, 1766; Samouelle, 1819)
The Scarlet Admiral (Harris, 1775b; Lewin, 1795)
The Alderman (Dutfield (2), 1748; Rennie 1832; Morris (2), 1853; Newman & Leeds (2), 1913)
The Red Admiral (Donovan, 1799: and most subsequent authors)
If Kaspar von Greyerz is correct, there were no widely used common names for Red Admiral and White Admiral butterflies prior to 1627; and to judge from Michael Salmon's book, even the name Admiral did not come into use (in reference to what later became known as the Red Admiral) until around 1699. Before that, people probably referred to that particular species of butterfly as some variant of "that butterfly with the red patches on its wings."
In short, "Admiral" (1699) appears to have been the first recorded common name for the Red Admiral butterfly, and "White Leghorn Admiral" (1703) appears to have been the first recorded common name for the White Admiral butterfly.