This result has nothing special to do with compact Lie groups: it works for
arbitrary topological groups $G$. And as Karol gently points out, the "expected
$G$-homotopy extension property" actually means "retracts of relative $G$-cell
complexes". The fibrations, like the weak equivalences are created by the fixed
point functors: $p$ is a $G$-fibration iff $p^H$ is a nonequivariant Serre fibration
for all closed subgroups $H$.

This is the ``Quillen model structure''; it was understood in the 1980's. There is
also a classical or Hurewicz model structure whose weak equivalences are the $G$-homotopy
equivalences and whose $G$-fibrations are the Hurewicz $G$-fibrations, defined by the $G$-CHP, and whose $G$-cofibrations really do have the $G$-HEP, which I imagine may be what the question had in mind. There is also a mixed model structure in which the weak equivalences are those of the Quillen model structure and the fibrations those of the Hurewicz model structure. Its cofibrations are the Hurewicz cofibrations that factor as composites of Quillen cofibrations and $G$-homotopy equivalences.

The proofs are really no different than in the nonequivariant case. Some references on my web page are http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf (Section VI.5), http://www.math.uchicago.edu/~may/PAPERS/MMMFinal.pdf (Theorem 1.8 there gives the same result as Schwede's A.1.18, with a little more detail about the properties of the Quillen model structure), and http://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf (Chapter 17)