Its only possible to have 1 clock master.
Best is to take the device as master, which is connected to the PC, in your case the HDSPe AES card.

It has operational advantages: if you change sample frequency in your DAW for your current project,
then all the other digital devices behind the HDSPe AES can learn the new sample rate automatically.

So make HDSPe to clock master (clock mode: internal)
The other devices you need to configure that they learn the clock from the proper digital IN port.
Then the ADI-8 DS MK III learns via AES from HDSPe AES and
the Tascam learns clock from ADI-8 DS MK III (you didn't tell how it is connected, whether you use AES or ADAT/SPDIF).

As a clock signal is only being sent via digital outputs, you have to connect
HDSPe AES "OUT" to ADI-8 DS MKIII "IN" and
ADI-8 DS MKIII "OUT" to Tascam "IN".

You didn't understand correctly. You always need a (one) clock master....

And then its usually best to choose the device close to the PC as I said.

> Now I'm just wondering if connecting the 3 RME clocks technically is better, worse or the same.

There can be only 1 clock master, but all other devices which are digitally connected need to
be synchronized by the same clock or you get phase problems.

> Perhaps the conversion load (time) is a fraction lower or higher with external word clocking,
> I don't know...I was hoping someone here would have such knowledge.

The clock has nothing to do with any form of conversion.
Its not a "load" time, its only the time until a clock circuit can lock on the master clock.
This happens very rapidly, especially RMEs steadyclock technology can perform this very quickly and reliable.

> Maybe it's no improvement at all to connect the 3 RME devices, and I can save the money on bying cables.

Again, you need to clock synchronize digital devices and you need one master for it.

And I would perform this by using AES or TOSLINK cables not having to connect the devices with an additional Word Clock cabling which adds some more cables to the setup ....

With RME devices its totally sufficient to use the digital data cables to transfer clock information.
One device is master clock, all the others are clock slaves and learn the clock from the master from a digital input
like AES, ADAT, SPDIF or MADI.

Between RME devices there is no high demand to implement word clock. Results can even be worse with WC, if very cheap devices have only a poor implementation of WC (like Behringer ADA8000, maybe also the successor). In these cases its better if such a device learns clock via its digital ADAT input.