However, if I click Help | About Update Services... it reports the version as 3.2.7600.226!
This question is addressed in SolarWinds KB4107.
Your server IS at build 262 (KB28281815 is installed).
Which still leaves us with the question of why your clients are not communicating.
The diagnostics for this will be found in the WindowsUpdate.log of a failing client.
Most notably, if these clients just became WSUS clients recently, they may have been updated to a YET NEWER Windows Update Agent via WU/MU, which will require that you install KB2938066 to the WSUS servers and PM servers. But before you do that, let's confirm with the WindowsUpdate.log