  The first observation of a gravitational wave (GW) and a short gamma-ray
burst (sGRB) emitted by the same binary neutron star (BNS) merger officially
opened the field of GW multi-messenger astronomy. In this paper, we define and
address \textit{lagging sirens}, a new class of multi-messenger BNSs for which
associated GWs and sGRBs are observed without the identification of their host
galaxy. We propose a new methodology to use the observed time delay of these
sources to constrain the speed of gravity that is, the propagation speed of
gravitational waves, the Hubble constant and the prompt time delay distribution
between GWs and sGRBs, even though a direct redshift estimation from the host
galaxy is unavailable. Our method exploits the intrinsic relation between GWs
and sGRBs observed and prompt time delays to obtain a statistical redshift
measure for the cosmological sources. We show that this technique can be used
to infer the Hubble constant at the $10\%$~level of precision with
future-generation GW detectors such as the Einstein Telescope and only 100
observations of this kind. The novel procedure that we propose has systematics
that differ completely from the ones of previous GW methods for cosmology.
Additionally, we demonstrate for the first time that the speed of gravity and
the distribution of the prompt time-delays between GWs and sGRBs can be
inferred conjointly with less than 10 sources even with current GW detector
sensitivities.
