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Due to the technical time delay of the XRT instrument on board the NeilGehrels Swift Observatory satellite, we cannot observe the X-ray emission lessthan $\sim40$~s after a gamma-ray burst (GRB) trigger. We here indicate a newstrategy of using the cosmological time dilatation in high redshift GRBs toobserve the earliest X-ray emission by Swift/XRT. We use $354$ GRBs with acosmological redshift from the Swift catalog, including short and long GRBs. Wefirst analyze the redshift distributions of the long GRBs of the differentbinary-driven hypernova (BdHN) families. We infer that the further evolution ofBdHNe II and III may be short GRB progenitors. We then compare and contrast thetime delay between the GRB trigger and the first observation by Swift/XRT,measured in the observer frame (OTD), and the corresponding delay measured inGRBs' cosmological rest-frame (RTD). We consider as prototypes three BdHNe I:GRB~090423 at $z=8.2$ with an RTD of $8.2$~s, GRB~090429B at $z\sim9.4$ with anRTD of $10.1$~s, GRB~220101A at $z=4.61$ with an RTD of $14.4$~s. This opens anew possibility for probing Episode (1) of BdHNe, linked to the newborn neutronstar ($\nu$NS) early appearance. In all three cases, we evidence a first regimerelated to the $\nu$NS spin-up by the supernova ejecta fallback and a secondregime leading to the $\nu$NS slowing down by the X-ray, optical, and radiosynchrotron emission. These two phases may be separated by a very shortgravitational wave emission due to a fast spinning $\nu$NS triaxialconfiguration.

Probing electromagnetic-gravitational wave emission coincidence in type I binary-driven hypernova family of long GRBs at very-high redshift