A Guild of Wizard Pupils?
 

Sauron was able to train the Mouth of Sauron with dark powers of sorcery to aid him where needed. Why then did the Elves, or indeed the Istari, not train and develop other races in wizardry?

The Mouth of Sauron was a human- a Black Numenorean. The Istari were actually Maiar in human form, whereas the Elves were born attuned to the Valar's power (magic).

J.R.R.T somewhat addressed this:

The Letters of J.R.R. Tolkien #155.

That would certainly seem to indicate that 'wizardry', as it describes the innate supernatural abilities of the Istari (and Sauron) cannot be taught to lesser beings.
At first glimpse, that might seem to be at odds with ROTK: We see the Lord of the Nazgûl

And the Mouth of Sauron, as mentioned

I think all this can be reconciled by noting that both the Witch-king and the Mouth had completely surrendered their sense of being and wills to Sauron. As such, they were essentially merely instruments of his will. Perhaps that allowed them the use of powers beyond their normal means under certain circumstances.

The Elves, to a much lesser extent, also had powers that appeared to Men to be 'magical'. Those also were abilities derived from the nature of their being and could not be transferred to others.

Yes, I think its more of a case of "you either have it or you don't". Unless you want to completely give yourself up, and become totally in the dominion of your overlord (in which case you a sort of an extension of them) then you can't gain magic. And then it isn't your own gain.

Look at the Drúedain. They can't even pass on their normal ("non-magical) tracking abilities to normal men:
So I don't really think that people could teach others "magic". Staying with the same theme, remember that the posssibly magical statues of the Drúedain were coveted by many?

Things like this just make me think that the people in Tolkien's world either had the power/magic (in which case it wasn't magical to them) or didn't (which is why they viewed it as "magic").

Now thats magic
 

I've wondered if Saruman had 'lab assistants' that he taught some basic explosives techniques to. Not magic per se, but things that would have been viewed as magic by the Rohirrim etc.

This is because the 'blasting fires of Orthanc' were used at Helm's Deep when Saruman was absent.

I'm hoping they were all called Igor and given to saying 'yeth marthter';)

Beorn seems to use his 'magic' as a natural ability, not learned in any way.

I've always believed the "Devilry of Saruman" used at Helm's Deep was indeed an explosive unknown to the Rohirrim or the Company of the Ring (save perhaps Gandalf). Gunpowder, maybe?

As for Beorn, I'm sure you're right, though he is apparently one of those "intentional enigmas" The Professor was seemingly so fond of.