  No-scale supergravity and the flipped SU(5)$\times$U(1) gauge group provide
an ambitious prototype string-inspired scenario for physics below the string
scale, which can accommodate the Starobinsky-like inflation favoured by
observation when the inflaton is associated with one of the singlet fields
associated with neutrino mass generation. During inflation, the vacuum remains
in the unbroken GUT phase, and GUT symmetry breaking occurs later when a field
with a flat direction (the flaton) acquires a vacuum expectation value.
Inflaton decay and the reheating process depend crucially on GUT symmetry
breaking, as decay channels open and close, depending on the value of the
flaton vacuum expectation value. Here, we consider the simultaneous
cosmological evolution of both the inflaton and flaton fields after inflation.
We distinguish weak, moderate and strong reheating regimes, and calculate in
each case the entropy produced as all fields settle to their global minima.
These three reheating scenarios differ in the value of a Yukawa coupling that
introduces mass mixing between the singlets and the ${\bf 10}$s of SU(5). The
dynamics of the GUT transition has an important impact on the production of
gravitinos, and we also discuss the pattern of neutrino masses we expect in
each of the three cases. Finally, we use recent CMB limits on neutrino masses
to constrain the reheating models, finding that neutrino masses and the
cosmological baryon asymmetry can both be explained if the reheating is strong.
