We calculate semileptonic form factors for the decays
$B_c \to \eta_c \, l \nu$ and $B_c \to J/\psi \, l \nu$
over the entire $q^2$ range, using
a highly improved lattice
quark action for charm at several lattice spacings down to $a=0.045$~fm.
We have two ways of treating the $b$ quark:
either with an $\O(\alpha_s)$ improved NRQCD formalism or
by extrapolating a heavy
mass $m_h$ to $m_b$ in the relativistic formalism.
Comparison of the two approaches
provides an important cross-check of methodologies in lattice QCD.
Nonperturbative renormalisation of the currents in the relativistic
theory also allows us then to fix NRQCD-charm normalisation
for $b$ to $c$ decays such as $B \to D$
and $B \to D^*$.