Why is the first integral Pontryagin class a homeomorphism invariant?

For spin manifolds this is proved in Corollary 1.22 (p.17) of Kammeyer's Diploma. Kreck's claim that the "spin" assumption can be dropped is mentioned after the corollary, with the caveat that "the author was unable to locate such a paper".

Incidentally, in dimension 4 the Pontryagin class (of a closed oriented smooth manifold) is proportional to the signature, and so is a homotopy invariant.

The topological invariance of the first Pontryagin class is proved in the paper

• B.L. Sharma. Topologically invariant integral characteristic classes. Topology Appl. 21 (1985), no. 2, 135–146. (link to Elsevier website)

In low degrees, Sharma computed the smallest multiples of the Pontryagin classes which are topological invariants, and Theorem 1.6 of his paper states that that multiple is 1 for the first Pontryagin class.

(Note: I found that via a discussion Section 22 of Rudyak's "Piecewise linear structures on manifolds", (arxiv link))