Which order is correct?
- National African weightlifting federations.
- African national weightlifting federations.
- African weightlifting national federations.
Simchona comments wisely that there is not a simple answer to this question.
A point specific to this actual question is that there is no nation of Africa; African, unlike American, is not generally used to apply to a specific country. South Africans (or at least those who use the expression) have hijacked the term in African National Congress, which is a shortening of the original: 'Members founded the organization as the South African Native National Congress (SANNC) on 8 January 1912' (Wikipedia). So the juxtaposing of 'national' and 'African' is questionable in the first place - are we referring to all the nations in Africa, to some of them, or to the organisations / practices ... that are nation-wide in the various countries?
Here, the order of the two words can be used to clarify what is meant (which takes precedence over ersatz-rules):
African national weightlifting federations (ie the national weightlifting federations to be found in the relevant African nations).
National African weightlifting federations (this implies that there is a special form of, or continent-wide association of, weightlifting, and we're referring to the local (ie within the different countries of Africa) organisations.
Compare Mexico / Mexican national American football team (or indeed United States national American football team). National Mexican American football team would be disastrous!
If you check with Wikipedia you would realize that the order in which prenominal adjectives and other modifiers of the noun are usually positioned is:
So the correct choice of expression would be "national African weightlifting federations". However, keep in mind that when it comes to naming something, a nation would tend to place its name at the start of the phrase, such as African National Congress, U.S. National Library of Medicine, etc.
The order of the adjectives in such matters has to do with specificity. With the above example consider the following questions:
The answer to the above is most likely 1. This means that we are talking about weightlifting federations.
Now, comparing the other two adjectives:
Again, we are very likely to be talking about 1.
This implies that the proper order would be African National Weightlifting Federations. This would represent federations at a country level throughout the continent of Africa.
(That said, I'm not sure if the use of federations by the OP is intentional. The usual choice is simply federation which would be an umbrella term used to group together a cluster of constituent bodies.)