OP the gain you will get varies from day to day but apart from the crappiest days you will get something
if the system is sized and installed correctly you can get cheaper hot water most of the year.
free - never
as you say the time you need the most water is in the morning this isn't a problem as you use the water heated the day before - topped up by the boiler
i have seen several installs like you describe and every landlord/hotelier i have met is happy with their system
use the solar to preheat and the boiler to top up in the morning
my own panel fitted on my old house was south facing (30 tube Apricus collector) it produced hot water without a boiler top up from may to oct (boiler wasn't even switched on) the cylinder was a 220 litre solar unvented cylinder fitted in the loft.
during the winter a 10 to 15 min top up in the morning produced enough hot water for daily use including an 8" rain water shower fitted in the bathroom
Holidays in Southampton for me this year then