Training regimen for the soldiers of the Diadochi... is there any record of one?

[Arjos]

Tbh saying "nobles that can ride well" is quite an understatement: these would've spent great part of their day training, both physically (gymnasion) and tactically...
While as Ludens pointed out, they were also klerouchoi, who would live in garrisons and train aswell...

If anything the hellenistic world had a greater number of professionals, compared to the roman one, during the early and middle republic...
This process started after the persian wars, when more and more poor Hellenes, devoted their lives to become mercenaries and later Makedones (picking up and developing boeotian tactics) trained both professionals and semi-professionals...
Just an example, here is Polybios on the preparations for Raphia's campaign by the Ptolemaioi:

Polycrates undertook the training of the cavalry of the guard, about seven hundred strong, and the Libyan and native Egyptian horse; all of whom, numbering about three thousand, were under his command. It was Echecrates the Thessalian who trained most admirably the cavalry from Greece and all the mercenary cavalry, and thus rendered most signal service in the battle itself, and Cnopias of Allaria too was second to none in the attention he paid to the force under him composed of three thousand Cretans, one thousand being Neocretans whom he placed under the command of Philo of Cnossus. They also armed in the Macedonian fashion three thousand Libyans under the command of Ammonius of Barce. The total native Egyptian force consisted of about twenty thousand heavy-armed men, and was commanded by Sosibius, and they had also collected a force of Thracians and Gauls, about four thousand of them from among settlers in Egypt and their descendants, and two thousand lately raised elsewhere.

Roma on the other hand, overshadowed them all in terms of manpower, greater flexibility in the field, not to mention political: most of the time it was socii/allies doing the dirty and hard work....