If there is any data, was the rate of admixture higher than today? Like WEHG brought the component in? Or was the amount maybe lower, like other groups like Early European Farmers or Yamnaya brought the admixture? Or was it evenly distributed?
tldr; Neanderthal ancestry is roughly equally the same between the groups you mentioned.
It's almost certain that the major pulse of Neanderthal admixture into non-Africans occurred very soon after the primary out of Africa event (here I am referring to the out of Africa migration which contributed towards most of the ancestry of all non-Africans today - of course there were other out of Africa migrations which were effective dead ends).
This is well established from a) looking at the size distribution of Neanderthal fragments in the human genome (ref, ref) and b) the fact admixture occurred around 232-430 generations before the 45kya year old UstIshim individual was alive. So most lines of evidence suggest this admixture took place between 50-60kya.
This is much earlier than the diversification of West Eurasian populations into groups like Yamnaya, Anatolian farmers and Western Hunter Gatherers. Therefore, there is no reason to expect any one of these ancestral groups would have had higher amounts of Neanderthal ancestry than any others, since the admixture event occurred into the common ancestor of all these populations.
Indeed, from Haak 2014:
The ancient individuals had indistinguishable levels of Neanderthal ancestry when compared to each other (∼2%) and to present-day Eurasians (Supplementary Information section 6)
There are some ancient individuals such as Oase1 who carried substantially more Neanderthal ancestry than modern day samples (in fact this individual has very recent Neanderthal relative), but that they didn't contribute anything towards the ancestry of later groups.
Btw I know a lot of these reference are old - I will add some more recent estimates later, but they don't change the overall picture.