So, my research is focused on the Bights of Benin and Biafra in the latter half of the nineteenth century, which means I have a passing familiarity with the Oudney-Clapperton-Denham expedition of 1822-23 and the trans-Saharan trade, but nothing especially intimate. I had actually meant to speak to a colleague of mine who studies climate in the near future about this precise subject but have not yet had the opportunity to do so in depth. Therefore, while I believe my command of the facts to be sufficient to answer your question, I should like to make clear to you that this response may potentially be incomplete.
As far as I know, academic literature on the climate of West Africa in the nineteenth century is extremely limited and none of it is written by historians – only climatologists. In fact, while there may be other pieces which touch upon the subject incidentally or attempt to determine the historical effects of nineteenth century climate patterns in the region, I am aware of only three articles dedicated to the subject of the climate itself:
Gallego et al. “An Instrumental Index of the West African Monsoon Back to the Nineteenth Century” Quarterly Journal of the Royal Meteorological Society Vol. 141 (2015): 3166-3176.
Nicholson et al. “Spatial Reconstruction of Semi-Quantitative Precipitation Fields over Africa during the Nineteenth Century from Documentary Evidence and Gauge Data” Quaternary Research Vol. 78 (2012): 13-23.
Villamayor et al. “Atlantic Control of the Late Nineteenth-Century Sahel Humid Period” Journal of Climate Vol. 31 (2018): 8225-8240.
I am uncertain as to why this should be the case, but I strongly suspect that part of the reason lies in the fact that there is simply very little data with which to work. As Nicholson et al. observe in their work, there are numerous and valuable sources of information to be had in the form of “settlers’ diaries, explorers’ journals, reports of scientific expeditions, historical chronicles documenting the reign of centuries of kings, oral tradition, missionary reports, and government records.” Yet, even when the documentary evidence is especially thorough, as with the reports of Heinrich Barth or the various missionary stations found in nineteenth century Africa, it is still part of an overall meteorological records which, “is largely fragmentary, especially prior to the 1890s, as well as qualitative.”
Gallego et al., for example, do not attempt to catalog contemporary precipitation data, instead employing the database produced by Nicholson et al. as a means of checking the accuracy of their preferred proxy - wind direction. They attempt to infer the length and intensity of the annual West African monsoon, based upon the wind directions recorded aboard ships off the West African coast and compiled in the International Comprehensive Ocean-Atmosphere Data Set. Villamayor et al., though including the precipitation data collated by Nicholson et al., relies upon the Hadley Centre Sea Ice and Sea Surface Temperature data set, and other data sets, to supplement and refine their climate model of the nineteenth century Sahel. I claim no expertise in climatology – and would be glad of correction from anyone trained in that field – but it seems clear in reading these articles that, though it is surely best practice to employ multiple data sets to guard against error, precipitation measurements simply do not exist in the quantities we would desire and a great deal of stock must here be placed in proxy measures.
Yet, limited though the records may be, and reliant upon inference as the academic literature is in attempting to extrapolate from those records, all three of the studies I have cited are in agreement on one key point: West Africa, the Sahara included, was more humid and more wet in the nineteenth century. The precise duration and intensity of this humid period, as well as the exact geographic extent of the effect, is still debatable, as each article comes to a different conclusion on those matters, but each agrees that all signs point to its having happened.
This would appear to be supported, anecdotally, in the text of other nineteenth-century travellers through the region such as Heinrich Barth. In Vol. I of his Travels (1857), for example, Barth asserts on page 383 that in the mountains north of Aïr, modern Agadez, “Here, indeed, a really tropical profusion of vegetation covered the whole bottom of the valley…”. On page 385 he reports of a nearby site that, “This valley, as well as those succeeding it, is able to produce not only millet, but even wheat, wine, and dates, with almost every species of vegetable; and there are said to be fifty garden fields (go-naki) near the village of I’farghen.” These descriptions are, in fact, fairly commonplace throughout the first volume, which covers the first leg of the journey south from Tripoli.
This is not to imply that the Sahara was, by any stretch of the imagination, Edenic. Barth nearly died several times as a result of exposure either to the climate. The desert was still a desert. Even were it not, there were other sources of danger ranging from the various diseases and parasites endemic in the region to war and banditry, both very real concerns in the interior. Nonetheless, given the limited academic research performed on the subject and the descriptions provided by travelers such as Barth, we may reach the exceedingly tentative conclusion that increased rainfall and humidity permitted a limited flourishing of plant life in the wadis, valleys, and other low-lying locations where water was liable to collect and which were partially protected from the full effect of the sun. Further research is required to confirm this conclusion and, if confirmed, to nuance it. However, while climatologists may find fault with the methodologies employed in the articles I have cited above, I can discern no basis for objection to their general conclusion.
Edit: Added clause "of the climate itself" to the second paragraph.
Edit No. 2: Substituted "climate of West Africa" for "climate of the Sahara" in the second paragraph.