Both are grammatical, but they have different meanings, one of which is unlikely to be possible.
Let's look at your sentence first.
When they finish, those old new-made materials will be handed out.
This one doesn't really make sense to me. New-made as a compound adjective means newly made or fresh. So the materials have the quality of being old and at the same time are also newly made. I think that is not quite possible.
When they finish, those old materials made new will be handed out.
This, on the other hand, makes sense. made here is a passive participle.
A participle is used to modify a noun. An active participle makes the noun it modifies an agent (the doer of an action). A passive participle, like this made, makes the noun a patient (the receiver of an action).
An active participle is in Verbing form, e.g. doing, running, touching; a passive one is in the past participle form, e.g. worn (wear), begun (begin), born (bear).
So, the materials in the sentence are the same old materials, but they will be improved, updated, and will be made new. And when they have been made new, they will be handed out.
Some people consider a participial phrase a 'reduced relative clause', which means the phrase is reduced from a clause similar to the one below.
When they finish, those old materials which will be made new will be handed out.
When they finish, those old materials
which will be made new will be handed out.
However, it is worth noting that this method doesn't always work well with sentences with participles, especially ones with active participles.
Anyone touching that wire will get a shock.
(taken from Practical English Usage, entry 411.2)
doesn't necessarily come from
who is touching that wire will get a shock.
It may be from
who touches touching that wire will get a shock.
or other sentences with different tenses.