Is red hair Celtic?

Note: The following article needs to be updated with information regarding origin of red hair and origin of today’s Celts going back further towards the ancient population of the Yamnaya.

Not too long ago have I asked if rh negative blood is Celtic and of
course the conclusion was that we cannot simplify such thing… or could
we? R1b is known as the Celtic marker and now that I have gone back to
Eupedia to look up the origin of red hair, I have found the following:

Although red hair is an almost exclusively northern and central
European phenomenon, isolated cases have also been found in the Middle
East, Central Asia (notably among the Tajiks), as well as in some of
the Tarim mummies from Xinjiang, in north-western China. The Udmurts,
an Uralic tribe living in the northern Volga basin of Russia, between
Kazan and Perm, are the only non-Western Europeans to have a high
incidence of red hair (over 10%). So what do all these people have in
common? Surely the Udmurts and Tajiks aren’t Celts, nor Germans. Yet,
as we will see, all these people share a common ancestry that can be
traced back to a single Y-chromosomal haplogroup: R1b.

The origins of haplogroup R1b are complex, and shrouded in controversy
to this day. The present author favours the theory of a Middle Eastern
origin (a point upon which very few population geneticists disagree)
followed by a migration to the North Caucasus and Pontic Steppe,
serving as a starting point for a Bronze-age invasion of the Balkans,
then Central and Western Europe. This theory also happens to be the
only one that explains the presence of red hair among the Udmurts,
Central Asians and Tarim mummies.

But what about the claim that red hair could come from Neanderthals?

If the mutation for red hair was inherited from Neanderthal, it would
have been from a Central Asian Neanderthal, perhaps from modern
Uzbekistan, or an East Anatolian/Mesopotamian one. The mutation
probably passed on to some other (extinct?) lineages for a few
millennia, before being inherited by the R1b tribe. Otherwise, it
could also have arisen independently among R1b people as late as the
Neolithic period (but no later).

More here:
The genetic causes, ethnic origins and history of red hair