How do prions cross the blood brain barrier?
I am not sure if a perfect answer is possible owing to the fact that Prion diseases are still incompletely understood, and are actively under research. On a preliminary search, I found some papers which address the issue (cited at the end).


Since $PrP^C$ is normally found in the body, its role and its transport is also relevant to the issue. It seems to undergo bidirectional transport across the BBB using a saturable mechanism, mostly a specific transporter (1). It is taken up peripherally, and by several regions of the brain with the possibility that it is involved in various brain functions including modulating the BBB itself. Amyloid Beta-protein, with several kinetic and dynamic similarities with $PrP^C$, is shown to have BBB altering effects. (2) How far do the pathological prions follow this is seemingly unknown.
Autonomic nerves seem to play a role in their transport.(3) When challenged with prions, sympathectomy delays or even prevents the disease, whereas sympathetic hyperinnervation accelerates prion pathogenesis. There have been studies to elucidate the transport via the nerves, such as (4) and (5).


There are several other studies on this issue, and following the cited references might help you find them. 

Reference number 3 also has information on the immunological response to prions.

References:


Banks WA, Robinson SM, Diaz-Espinoza R, Urayama A, Soto C. Transport of prion protein across the blood–brain barrier. Experimental neurology. 2009 Jul 31;218(1):162-7.
Giri R, Shen Y, Stins M, Du Yan S, Schmidt AM, Stern D, Kim KS, Zlokovic B, Kalra VK: Am J Physiol Cell Physiol. 2000 Dec; 279(6):C1772-81.
Aguzzi A, Zhu C. Five questions on prion diseases. PLoS Pathog. 2012 May 3;8(5):e1002651.
McBride PA, Schulz-Schaeffer WJ, Donaldson M, Bruce M, Diringer H, Kretzschmar HA, Beekes M. Early spread of scrapie from the gastrointestinal tract to the central nervous system involves autonomic fibers of the splanchnic and vagus nerves. Journal of virology. 2001 Oct 1;75(19):9320-7.
Glatzel M, Heppner FL, Albers KM, Aguzzi A. Sympathetic innervation of lymphoreticular organs is rate limiting for prion neuroinvasion. Neuron. 2001 Jul 19;31(1):25-34.