There is no Rules-As-Written answer. This is not discussed in either the Player's Handbook or the Dungeon Master's Guide, in either the 2nd Edition or 2nd Edition Revised books.
Weapon and Non-Weapon Proficiency rules were optional in 2nd Edition. My guess is that because of this, the writers didn't think through every possible scenario. But that's just speculation on my part.
So, we're left to muddle through with our own common sense. If you google, you will find discussions about this, with varying opinions. This is the method I used the only time it became relevant in my 2nd Edition GMing:
- Weapon Proficiency: Gain the level 1 proficiency slots for the new
class, and advance as the new class. This of course includes using
the new class' level 1 THAC0.
- Non Weapon Proficiency: No new NWP at level one. However, I allowed the character to
1) use existing NWP without penalty
2) regain one NWP slot he'd previously paid as a penalty to learn an out of class NWP which was now in-class for the new class.
3) Spend the regained NWP immediately on a new class NWP
4) Earn NWP per the progression for his new class.
My main goal was to get the bookkeeping over with so we could get back to our game. The player wasn't happy with my decision about NWP, but I couldn't really justify his character suddenly knowing three new things, especially in his specific case where the decision to multiclass (From priest to fighter) came pretty spur of the moment as a result of in-game actions, and not through any long-term planning. The player agreed that did make more sense, but still wanted the extra NWP. I was not persuaded by that argument.