In Section 3 of this note of Haines, it is proved that the fibers of the Bott-Samelson resolution have a paving by affine spaces. This seems to be reasonably explicit to investigate the structure, compute cohomology and things like these.

There are also the two papers of Stéphane Gaussent on the fiber of the Bott-Samelson resolution:

- The fibre of the Bott-Samelson resolution, Indag. Math. (N.S.) 12 (2001), No. 4 453-468
- Corrections and new results on: "The fibre of the Bott-Samelson resolution" [Indag. Math. (N.S.) 12 (2001), no. 4, 453--468; MR1908873], Indag. Math. (N.S.) 14 (2003), No. 1, 31-33

I think statements like the cell decomposition do not depend on the base field. This is at least true for the Grassmannians, which exist as schemes over the integers, such that the usual cell decomposition does not depend on the choice of base field.

I am not sure if this is what you are looking for, and maybe you already know about the above references...