RAW: Ambiguous
The rules are not clear on what, exactly, constitutes an “arcane spellcasting class” or a “divine spellcasting class,” probably because initially it was obvious. By default, assassins, bards, sorcerers, and wizards were the arcane spellcasting classes, blackguards, clerics, druids, and paladins were the divine spellcasting classes. There was no ambiguity: the former are the classes that cast arcane spells, and the latter are the classes that cast divine spells. Easy.
Then they printed things like Alternate Source Spell, Rainbow Servant, Sha’ir, and Southern Magician, which introduced ambiguity that hadn’t existed before. In these cases, you have spell slots from one class being used for either arcane or divine spells.
Does this new feature suddenly make the class into an “arcane spellcasting class” or “divine spellcasting class” where it wasn’t before? There hadn’t ever been a strict definition before, and they didn’t print one at this point, either. Some of these effects seem to try to include wording that prevents this kind of thing, but much of those rules are also unclear. For instance, consider this from Southern Magician: “The actual source of the spell's power doesn't change,” which Customer Service interpreted as preventing entry to mystic theurge. But it doesn’t really say that, does it? It says something about power source, which is unclear.
Unfortunately, there’s no direct, rules-as-written, “as it says on page xyz of Complete Shenanigans” kind of answer to this question.
Recommendation: Never
RAW is ambiguous, but what’s going to work well in-game is not: never, under any circumstances, should one be allowed to advance wizard spellcasting faster than the wizard does. That should never, ever happen in any game, and if you’re going to allow it you might as well allow Pun-pun.
Allowing these sorts of tricks to qualify for mystic theurge, and other prestige classes and feats that require one type of spellcasting or the other, is pretty clearly legal, RAW, and also usually far less troublesome. The only exception I’d be likely to make is the dweormerkeeper from Complete Divine’s web enhancement, but then I’d probably just ban that class outright.
Even allowing a divine-only prestige class to progress wizard spellcasting is almost-always not a problem. It’s the double-progression that should never, ever happen.
Skill Point Cost: Only current class’s class skills
From Skills Summary
Each skill point you spend on a cross-class skill gets your character ½ rank in that skill. Cross-class skills are skills not found on your character’s class skill list.
From Player’s Handbook pg. 60.
Skill points must be spent according to the class that the multiclass character just advanced in.
Max Ranks: Once a class skill, always a class skill
From Skills Summary
Regardless of whether a skill is purchased as a class skill or a cross-class skill, if it is a class skill for any of your classes, your maximum rank equals your total character level + 3.
From Multiclass Characters
If a skill is a class skill for any of a multiclass character’s classes, then character level determines a skill’s maximum rank. (The maximum rank for a class skill is 3 + character level.)
If a skill is not a class skill for any of a multiclass character’s classes, the maximum rank for that skill is one-half the maximum for a class skill.
Best Answer
No
Unless the prestige class is from Book of Exalted Deeds, Complete Warrior, or Complete Arcane, which specify that those prestige classes do.
The blurbs written in Complete Warrior and Complete Arcane are written as if they were establishing a global rule, but this contradicts the rules for prestige classes in the Dungeon Master's Guide, which mentions no such rule and only states that requirements are necessary to take one’s first level of a prestige class. After the first level has been taken, according to the Dungeon Master’s Guide, all bets are off. You can lose the requirement, and you not only keep the class features, you also retain the right to continue taking levels.
In fact, the Dungeon Master’s Guide includes a prestige class which breaks if this rule is applied: the dragon disciple. The dragon disciple’s requirements include being a non-dragon race, but the capstone of the class applies the half-dragon template, which includes changing the character into a dragon—and thus making it impossible to retain his or her qualification for the class. Under the real rules, this is fine, but under Complete Warrior/Complete Arcane mistakenly applied globally, we end up with Schrödinger’s dragon: gaining the class feature causes the character to lose dragon disciple class features, including the one that broke the requirements, so he or she qualifies again, and then regains the class features and thus breaks them again!
But this is not a problem, because the system includes rules for handling such contradictions.
The errata rules state that when two books contradict each other, the “primary source” wins; Complete Warrior and Complete Arcane are the primary sources for the prestige classes in that book, but the primary rules for the very concept of prestige classes in general is the DMG. So those books can establish an exception for themselves, but not globally. To do that would require a mention in the DMG errata file, which despite reprintings since Complete Warrior and Complete Arcane, has not happened. The DMG was not the only book to print prestige classes that play badly (as in, don’t make sense) with this rule, either.
In contrast, on this subject, Book of Exalted Deeds is written “correctly.” It explicitly supplies an extra rule for its prestige classes (page 49), without trying to assert that such a rule applies to other prestige classes in other books.
In general, gaining prerequisites temporarily is a fairly high-optimization trick; it's probably not welcome at a lot of tables. This is typically more a matter of gentlemen’s agreement than house rule, though.
The reason not to establish this as a rule is because, if, in combat or something, someone is drained or cursed and loses prerequisites, it is not appropriate for every prestige class to have a pseudo “falling” mechanic. The Book of Exalted Deeds prestige classes can explicitly fall and that’s appropriate1 to that book, but it’s not appropriate for all prestige classes everywhere. It’s better to just say that you don't want people building characters around this trick.
Finally, do consider that the majority of prestige classes have requirements more difficult and painful to meet than they deserve, especially warrior classes which tend to be for the weakest base classes. Thus, if someone is interested in a trick like this, the correct response may easily be: “I don’t want shenanigans like that in this game, but how about we just waive that prereq entirely?” Then everyone’s happy.