Let’s make the puzzle a little easier first: let’s say it’s enough if all coins are the same side up. It doesn’t matter if heads or tails.

If all coins are turned same side up we are done.

Wiseman will assume that two coins are heads up and two coins are tails up.

He turns two coins in opposite corners. If his assumption was correct and two same side up turned coins were not next to each other (were across from each other) warlock will end the game now.

If it did not happen wise man will not give up his assumption about two + two coins yet. If two coins weren’t across from each other they have to be next to each other (the previous turn didn’t change it).

Wiseman will turn two random coins placed next to each other which should (if the 2+2 assumption is correct) either end the game or the coins will be across from each other. If it didn’t end the game wise man will simply turn the coins in opposite corners and that should end the game.

Game is still not over? It seems that wise man’s assumption about two and two coins was wrong. It means that one coin is facing up different side than others. Now it’s easy. Wiseman turns any coin. Now they are either all the same or two and two.

It means that wise man will repeat what was described above. Now he is sure that there is heads on two coins and tails on the other two.

Let’s get back to the original puzzle. It is a bit more complicated but just a little bit. Wiseman does the same as described above only his every other turn will be turning all four coins.

I hope I explained it clearly enough.

Answer:
The moves are:
1) turn two coins if opposite corners
2) two random coins placed next to each
3) same as 1
4) turn any coin
5) same as 1
6) same as 2
7) same as 1
8) flip all coins
9-15) same as 1-7

Explanation: moves 1-7 will cause the coins to be all head or all tails (where 7 is the worst case),  but if after the 7th step the game is not over yet that means we started the game with all coins as tails or we got an all tails
situation during the steps. so that’s why on the 8th step we flip all the coins and repeat the process. (The worst case is indeed when we  start with all coins as tails!!)

Notice that the moves 1,2,3,5,6,7 doesn’t change the number of pair of coins already is head position. move 4 does.
Now if at beginning we had 2 heads than if thy were diagonal than after step on we get all heads or all tails. If they are next to each other than step 2 will cause them to became diagonal and step 3 will bring us to the same result as step 1.
But this will not work is the initial state was even number of heads/tails. So that’s why on step 4 we turn only 1 coin. So now we are sure that we have 2 tails and 2 heads so we repeat steps 1-3.
In all those cases (2,2 or 1,3) the worst case will be that only on the 7th step we get all tails or all heads and so step 8 will stop the game if we got the all tails situation.
But if we started with all the coins as tails we will need 7 more steps.