Apologies if this sounds familiar! We have a downstairs electric shower which is gravity fed direct from the mains. Problem is the shower runs cold if the loo is flushed or another tap is turned on. Annoying for anyone who happens to be in the shower at the time. Even if not in the shower, you have to wait for the pressure to return. I think this may be due to the work that has been done with the piping over the years, which is probably not practical to replace. Would a power shower solve this problem? Extending the hot supply to the shower would be a bit messy but that is do-able at a cost.
Just to clear one thing up, Johnny........ mains water is at mains pressure, not gravity. That would come from a roof tank.
Anyway, bringing in a hot supply would only work if you have a combi boiler or a mains pressure hot water system. A tank-fed gravity hot would be out of balance with a mains pressure cold.
Gravity fed supplies can be greatly affected by poorly thought-out alterations to pipework. Mains pressure is not usually affected in most cases. What alterations do affect is the "flow", that's the actual amount of water flowing rather than the pressure.
It does sound as though you have very low pressure in your mains. Are you on high ground?
What's the pressure like at the kitchen sink? (Assuming the mains comes in there.)
Feeding the cold supply from the mains as near a point where the flow enters the house may help.
If you are obliged to have a separate cold tank, maybe it'd suit the shower to have a tank for itself. Failing that a separate pipe from the existing tank, unassociated with any loo feed.