It seems that the LNPs (lipid nanoparticles) that contain the mRNA of Covid-19 vaccines from BioNTech and Moderna do - at low doses - pass the BBB. This is mentioned by the EMA several times in their report, for example p. 54 and discussed in the comments of an article on Derek Lowe's blog.
If that's indeed the case, what would happen once the mRNA + nanolipid reach the brain? Which cells would pick up the LNPs and for how long would they stay in the brain? If there is cells that can transform this mRNA in proteins, where will these proteins then go, and for how long will they stay in the brain? What about the LNPs: what can/will the brain do with the remaining lipids?
More readings: