How to define & differentiate, "on hand", "to hand", "at hand"?

You could say that, if for all these four locutions, you keep to the meanings that have to do with having something at one's disposal (and that seems to be the idea in this concern for differentiating them from one another), then there is a similarity between "on" and "in" as they connote availability, and between "at" and "to" as they connote the idea of being easily available. However, the particular idiomatic use of each will show that much finer distinctions are to be made, and that this rough guideline is insufficient to warrant a proper use in each given context (for instance, "on" is preferable to "in" in context where providing help is the matter at hand).
There is then a difference that can be made between "at" and "to": whereas the former connotes easiness because of nearness (in space or time), the latter connotes easiness of a more general nature; this is reflected in the fact that the adverb "close" is more often used to reinforce "at hand" (ngram) and the adverb "ready" more often used to reinforce "to hand" (ngram). I suppose that we can say that when the context calls specifically for "at" you can be less precise and use "to" without being significantly off the mark.

The definitions of these phrases can be listed for comparison (SOED, examples from OALD);

in hand
​in actual possession, to one's disposal; to spare
♦ if you have time or money in hand, it is left and available to be used (OALD)
♦ We managed to redecorate the house and still have some savings in hand.
♦ She completed the first part of the exam with over an hour in hand.

on hand
in one's possession, in one's charge or keeping
available, especially to help (OALD)
♦ The emergency services were on hand with medical advice. (OALD)

at hand
a) near, close by b) near in time
♦ There are good cafes and a restaurant close at hand. (OALD)
♦ Help was at hand. (OALD, 2005 paper ed.)
Note: this expression is reinforced often by the adverb "close": "close at hand".

to hand
within reach, accessible
♦ I'm afraid I don't have the latest figures to hand. (OALD)
Note: this expression is reinforced often by the adverb "ready": "ready to hand".