How long does it take until the holding tanks of those reactors are full and what will they do if there is still no processing plant built anywhere?

We have heard the nuclear argument before. We know what is missing. Nothing has changed with regard to processing and disposal.

Dry cask storage could hold spent fuel for decades, and it's only a few centuries before it's less radioactive than uranium ore.

Storage would be considerably less complicated due to the properties of thorium.  Th-232 captures a neutron and (through two beta decays) becomes U-233.  It takes another FOUR neutron captures to turn U-233 into U-237, which can decay into neptunium; another neutron capture is required to create Np-238 which decays to plutonium.  To get there, all the intermediates have to avoid being fissioned by any of the neutrons.  The consequence is that the spent fuel of a thorium reactor would have only minuscule amounts of plutonium.