Restorers say that if allowed to remove and replace the discolored

Dear RichaChampion,
I'm happy to respond. My Magoosh job affords me a little time to attend to GMAT Club.

The word "if" is a subordinate conjunction, a word that begins a subordinate clause. It must have a full clause after it, BUT sometimes that clause can be implied rather than explicitly stated. Thus, technically, (A) is not grammatically wrong, because it implies the subject/verb pair (... say that if [they are] allowed to ....). It is simply too casual. It doesn't sound formal.

You are correct that (B) change the number, and (C) of course is correct.

(D)-(E) contain the "when" split about which you are asking.

The word "when" is another subordinate conjunction. The difference is in connotation. Consider these two statements.
1) I will give my sister a gift if she visits me.
2) I will give my sister a gift when she visits me.
These are both grammatically correct but have different connotations. The first implies that the visit of my sister is uncertain: she may or may not visit, and I don't know which will happen. If she does visit, then I have a gift for her, but it's not clear at all what will happen to this gift if my sister doesn't visit. Statement #1 is a statement full of uncertainty about how the future will play out.
By contrast, in #2, we confidently can infer two separate facts about the future. It is confidently asserted that my sister will visit: that is not in doubt. It is also confidently assert that, when she is here, I will give her a gift: that is not in doubt either. Statement #2 is a statement of crystal clear certainty.

In much the same way, (A)-(C) imply that we do not know whether the restorers will be "allowed to remove and replace the discolored layer of varnish on the Mona Lisa." There is uncertainty about the future in the "if" construction. The restorers may or may not ever receive this permission, and these changes to the painting may or may not ever take place. We have no idea how it will turn out. Nothing in sentence gives us any insight into how things will unfold.

The "when" construction in (D)-(E) replace the uncertainty with utter certainty. This version implies that we know for a fact that the restorer will be allowed at some point to make these changes, and that the painting definitely will be changed in the manner described. This is a HUGE change in meaning, a change from uncertainty about the future to complete certainty about it.

Does all this make sense?
Mike