Timeline of Himalayas/Tibetan plateau formation
Drying out of North Africa is likely the result of something far more local. I've never heard of the either the Tibetan Plateau or the growing Himalaya as causing changes to Africa's monsoons and always understood the connection between Milankovich cycles and the consequences of glaciation. According to this study, which modelled weather patterns of the time:

Aridification of the Sahara desert caused by Tethys Sea shrinkage during the Late Miocene

The drying of Africa started much earlier and was likely related to orbital cycles of glaciation-induced drying of the Mediterranean region.


  Through a set of experiments, we demonstrate that the African summer monsoon was drastically weakened by Tethys Sea shrinkage, allowing arid, desert conditions to expand across north Africa. Not only did the Tethys shrinkage alter the mean climate of the region, it also enhanced the sensitivity of the African monsoon to orbital forcing, which subsequently became the major driver of Sahara extent fluctuations.