I can now confirm that an old Hughes sat phone, on Thuraya, works fine in the cockpit.
The data rate varies 0.8 to 1.3 kbytes/sec depending on how good the signal is.
A direct line of sight (no in-cockpit obstructions) is essential - possible only if flying a relatively constant heading. So the sat phone will need to be moved about accordingly. It cannot lie down; the antenna must be vertical.
No speed related effects were seen up to 185kt ground speed, on any heading, so doppler shifts (worst case to/from the satellite) don't seem to matter.
This was an old phone, 7100 I think, and I will next try a newer one.
It takes about 45-60 secs to establish a dial-up connection.
Thuraya also block VPN access, which is pretty common on mobile networks; Voda are the only exception I know of. I dare say a VPN on port 80 or 443 (HTTP) would work fine... not that this will bother many pilots since the main use of this would be for simple text only sites for tafs/metars and possibly meteox.com.
The plane was a TB20GT; composite roof. However only the windows were signal transparent.