The Galileo spacecraft traveled a tortuous path through the inner solar system in order to get to Jupiter. It was launched from the Atlantis Space Shuttle, STS-34 at 7:15 PM, EST on October 18, 1989 and its path took it by Venus in February, 1990; Earth on December 1990, Earth again on December 1992 and finally it arrived at Jupiter on December 7, 1995. In 1992, its predicted arrival was at 14:04 PST which was, so far as I have heard, identical with its actual arrival time to within 60 seconds.
On December 7, 1995 Jupiter was 6.24 Astronomical Units or 580 million miles from the Earth, and the light travel time was some 3110 seconds or 51.9 minutes to Earth. So, there was an unavoidable time lag of 51.9 minutes between Galileo sending a message and Earth astronomers receiving it. As Galileo dropped off the Probe and continued its orbital insertion maneuver, Earth was occulted by Jupiter for about 3.5 hours, but this was also known well in advance. Because of the problems with the main antenna, Galileo was forced to store about 37 minutes of data from the Probe prior to transmitting to Earth, but this data is fully recoverable, however, at the expense of other data that is now completely lost to astronomers.
There was, however, an unexplained and rather mysterious 53-second delay at the start of the data transmission. This means that the Probe got deeper into the Jovian atmosphere than originally had been planned before data taking began. The reason for this delay is not known. Both the Probe and Galileo had to navigate intense radiation belts a thousand time more lethal than the Earth's Van Allen radiation belts. Perhaps a stray charged particle impact on the delicate electronics produced a momentary software glitch. We may never know the exact reason for this. Despite the enormous speeds of the spacecraft, the delay could not have been any form of relativistic 'time dilation', nor is Jupiter massive enough for the delay to have been produced by the so-called gravitational redshift process.