The real-world Druids of Gaul used the Massilian Greek and Inscriptional Latin alphabets. The Druids of Ireland used the Ogham "tree-letters" script. The Druids of Britain may've used both; though in modern times, a Bardic script was retroactively attributed to them.
There is a specifically Irish precedent in the name of the Shillelagh spell, a Gaelic word. Also, the AD&D 1st Edition bardic schools were based on the Irish Gaelic language. And 1st edition Bards had to dual class as a Druid in order to become a Bard.
Since Ogham is more quintessentially 'Celtic' than Greek alphabet, I would suggest using Ogham to represent Druidic script. (And Classical Irish to represent the Druidic language.)
Edit: Above, I gave at least three in-game examples of Irish Gaelic language being used to represent D&D druidic terminology. Yes, that is not fully explicit 'proof' of D&D Druidic script equating to Irish Gaelic Ogham script, but it certainly is implicitly related.
Here are two more implicit suggestions that Celtic parallel may be especially relevant in filling out the hinted details of the D&D Druid and its script:
WotC officially confirms the Celtic 'origins' of the D&D druid, in WotC's official product history of the AD&D2E Celts Campaign Sourcebook:
"D&D is a hodge-podge of different historical and fantastic influences; the Celts entered that melting pot primarily thanks to B. Dennis Sustare, who provided the basics of the druid class that were fleshed out in Eldritch Wizardry (1976). Granted, the AD&D druid moved somewhat distant from its Celtic roots, but every once in a while an article in Dragon would remind D&D players of those origins, with Bill Fawcett's "The Druid in Fact and Fantasy" from The Dragon #32 (December 1979) being one of the best."
And Bill Fawcett's above-mentioned 1979 article "The Druid in Fact and Fantasy," published during the time of Gary Gygax's oversight of the magazine, further points to the Celticity of the AD&D Druid, as close parallels. In regard to scripts, see p.21:
"The Celtic culture was 'prehistoric' in that writing and literacy were virtually nonexistent. The Celts never did develop a written language that was universally used. Later the written languages of nearby cultures were adopted, particularly Latin after Caesar's conquests. Therefore, the tradition of Druidism was entirely oral. […] This is reflected by the inability of a Druid to use any written magical items. Presumably this includes all tomes, scrolls, and similar types of paraphernaila."
and the somewhat contradictory p.32 "Together with the Bards and Seers, they formed the priesthood and literate class of the Celts for the entire history of the culture."
Fawcett's quote hints at the written language that some Celts did develop, but which wasn't "universally used"—namely the Ogham developed by the Celts of Ireland. Obviously, Latin or Greek don't represent a 'secret script.' That leaves Ogham (or a 'D&Dized' version of Ogham).
Given the Irish Gaelic name of the Druid spell Shillelagh (see the Wikipedia article) the Irish Gaelic names of the AD&D Bardic colleges (Fochlucan, Mac-Fuirmidh, Doss, Canaith, Cli, Anstruth, Ollamh—with Ollamh being the real-world name for a high-ranked Bard or Druid in Ireland, and not in any other Celtic culture) and given that 1e Bards were all required to be Druids, and thus presumably from the same cultural milieu as the 1e Druid), it follows implicitly (but not explicitly), that the traditional, pre-moden Irish Celtic script--Ogham--is a fitting model for the secret script of the D&D Druid, at least in 1st Edition AD&D, if less explicitly in later, more 'hodge-podged' evolutions of the Druid. This is implicitly supported by the above facts, though not explicitly.