Skara Brae was occupied likely between roughly 3000-2500 BCE - approximately when the Great Pyramid was being constructed in Egypt for context, and is indeed one of the best preserved Neolithic villages in Europe. There hasn't been massive world wide climatic shifts since that time (of course there are smaller regional shifts), but I think what you are really asking is whether or not the site was abandoned due to climatic regions.
Really, this is a compelling question, because the site truly looks like you could move in tomorrow (as long as you installed a tarp for a roof!). The site is amazing because it is completely stone built, as opposed to contemporary mainland villages that would have been wood-built and have since decomposed into postholes and dirt stains - evocative to an archaeologist but not exactly exciting to the layman. Skara Brae also has stone-built furniture, something that we usually can only speculate about.
The buildings are surrounded by middens, full of animal bones like deer, cow, sheep, pig, fish, bird, whale, and many other species. They were clearly skilled and varied hunters and herders. In that respect, wild food supply was likely much more abundant 5000 years ago.
So it's very tempting to suggest that something dramatic must have happened in order to force the residents out of what is obviously such a well preserved and abundant lifestyle.
In fact, this was one of the earlier suggestions, that some sort of storm or changing climatic conditions must have forced the residents to relocate. However, more recent scholarship, has started to question that assumption since we see changes in lifestyle and villages throughout Neolithic Britain, with the start of construction of henges and other large-scale monuments. Some scholars have suggested that this indicates the emergence of a different social structure, revolving around the ability to mobilize labour, leading to different types of settlements - instead of close knit villages like Skara Brae, we see more dispersed settlements.
At Skara Brae in particular, there is also some indication that erosion was becoming a problem, with wind-blown sand starting to show up in the sediments around the village. This obviously might have been one reason why Skara Brae itself does not appear to have continued as a viable settlement location.
All this to say that Skara Brae is a beautifully evocative site since it is built of stone and therefore shows us all the aspects of living that would have been present at all sites, but have since decomposed. There was not any dramatic climatic shift that drove people away from the site, though erosion and subtle shifts might have contributed to what appears to be a full-scale reorganization of lifestyles that was occurring around 2500BCE.
Sources:
Childe, V. G., & Clarke, D. V. (1983). Skara Brae. HMSO. -The original excavation report, see all the lovely finds!
Bayliss, A., Marshall, P., Richards, C., & Whittle, A. (2017). Islands of history: The Late Neolithic timescape of Orkney. Antiquity, 91(359), 1171-1188.
Simpson, I. A., Guttmann, E. B., Cluett, J. and Shepherd, A. (2006), Characterizing anthropic sediments in north European Neolithic settlements: An assessment from Skara Brae, Orkney. Geoarchaeology, 21: 221–235
Jones, A. (1998). "Where Eagles Dare: Landscape, Animals and the Neolithic of Orkney" Journal of Material Culture 3: 301-324
You might try /r/archeology, too