I've changed suppliers quite a few times. It's not easy to check if it will save you money, but not that hard either.
You need to know how many kWh you use in a year and you need to know the standing charge. Then a spreadsheet, or calculator and back of a fag packet, will tell you the totals.
I have been with the cheapest suppliers for several years now, and so I have saved a bit of money, but only by being on the horse which for the time being is slightly trailing the pack. Over time they just switch positions, that's all, and customer inertia probably means the trailing horse keeps many of its customers when it surges towards the pacemakers. Over time, taken as a whole, energy supply is one of the greatest rip-offs and cons ever perpetrated on us as it just merrily careers upwards, impervious to wage or price inflation.
What would work is if EVERYBODY switched to the cheapest provider, so no other supplier suddenly had any customers at all. But obviously they count on this not happening, and on customers either not bothering to look, or thinking the savings aren't worth the inconvenience.