I have not read it (It is on my ever-growing todo list), but the paper
On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds.
J. Topol. Anal. 2 (2010), no. 4, 469–526.
58E05 (37D15 57R19)
should contain the proofs of what you want. From the Mathscinet review of D. Hurtubise
This paper contains precise statements and careful proofs of several
essential results that are fundamental to the moduli space approach to
Morse theory. Most of the results in this paper have appeared and/or
been used in other papers, but this is the first self-contained
reference that provides clear and complete proofs of all of the
following: (1) the smooth structures on the compactified spaces that
arise from the gradient flow of a Morse-Smale function, (2)
orientation formulas for the strata of the compactified spaces, and
(3) the CW structure determined by the unstable manifolds of a
Morse-Smale function. The results are proved for a Morse function on a
complete Hilbert manifold that satisfies the Palais-Smale Condition
(C) and has finite index at each critical point (the CF case).
Of course this also proves the CW structure in the finite dimensional case.