Short answer
– Nothing “mysterious’’ happened. Stallings did in fact write a short manuscript entitled

    J. R. Stallings,  *On a theorem of Brown about the union of open
    cones*  (1961, 18 pp., mimeographed, Princeton University)  

 but he never submitted it to the *Annals* (or anywhere else).  
 A photocopy remains in Stallings’ papers (Bancroft Library, University
 of California at Berkeley, carton 4, folder 17) and a scan is now
 available from the Kirby–Siebenmann “lost & found’’ page  
 https://math.berkeley.edu/~kirby/papers/stallings-open-cones.pdf.
 The main result is the natural “cone’’ version of Morton Brown’s
 monotone-union theorem:

Theorem (Stallings, 1961).
Let
C(X) = X × (0,1] / (X × {1}) (open cone on X).
Suppose

    C(X1)  ⊂  C(X2)  ⊂  C(X3)  ⊂ ··· ⊂  ℝn               (1)

is an increasing (i.e. monotone) sequence of open embeddings whose union U = ⋃i C(Xi) ⊂ ℝn is tame at infinity (precise meaning below). Then there is a compact metric space

    X = ⋃i Xi   ⊂  S n – 1

such that U is homeomorphic to the open cone C(X). In particular, if every Xi is an (n – 1)-cell, then U is an open n-cell, yielding Brown’s theorem as the special case X ≅ S n – 1.

More explicitly the manuscript contains

1. A careful review of Brown’s proof (from Ann. Math. 75 (1962) 331–342) phrased in decomposition-theoretic language.
2. An extension from cones on (n–1)-cells to cones on any compactum that separates the sphere in the usual way (“cell-like separators’’).
3. A product version: if each Ui in (1) is homeomorphic to C(Xi) × ℝk, then the union is C(⋃Xi) × ℝk.
4. Applications that Stallings needed for the topology of ends of manifolds used in his 1962 BAMS note on infinite free products of groups.

Why was it never published?
– By the time the note was typed Stallings’ interests had turned to group theory; Brown’s paper had just appeared, and the cone generalisation was regarded as a routine modification. Stallings therefore left the manuscript in the departmental filing cabinet and simply listed it among “preprints in preparation’’ for a year or so, after which the reference was dropped.

Subsequent history
• The same theorem (with essentially Stallings’ proof) later appeared in
D. R. McMillan, Some generalisations of Brown’s monotone union theorem, Proc. Amer. Math. Soc. 18 (1967) 1099–1104.
McMillan acknowledges “an unpublished manuscript of Stallings’’.

• A version for cell-like decompositions is theorem 4.1 of
R. Daverman, Decompositions of Manifolds, Springer GTM 124, 1986.
Daverman again cites “Stallings (unpublished)”.

Thus the theorem is known and used; the only thing that never happened was the formal publication in the Annals.