(1) is Théorème 7.4.18 of [[BT1]: Bruhat and Tits - Groupes réductifs sur un corps local](http://www.numdam.org/item?id=PMIHES_1972__41__5_0).

Now let $(A_1, C_1)$ and $(A_2, C_2)$ be two apartment–chamber pairs.  Upon replacing $(A_1, C_1)$ and $(A_2, C_2)$ by $G$-conjugates, we may, and do, assume that $A_1$ and $A_2$ both equal $A$.  Now the entire computation is happening with respect to a fixed affine root system, and has nothing to do with the full valuation of root datum, so we go back to the very beginning.  (1.3.3) of [BT1] shows (or, rather, quotes from Bourbaki the fact) that there is some element of $W$ that conjugates $C_1$ to $C_2$.  This element of $W$ may be lifted to an element of $N \subseteq G$.  Per your [definition](https://mathoverflow.net/questions/431445/does-an-affine-building-associated-to-a-group-satisfy-the-axioms-of-building/431467#comment1110480_431445), this is what is required in (2).