The fuel pump, filter and level sensor are in a can that sits in the tank. On a Jazz this is accessed through a hatch under the centre console rather than from outside the car. (Pic. attatched*) Whoever dismantled this can to change the pump has affected the level sensor.
An electrical fault would mean the gauge would either work or not, it would not be slow to respond. Looking at the wiring diagram, the fuel gauge along with the low fuel warning lamp (and the speedo & tacho etc.) are driven by a microprocessor within the dashboard unit. This takes an input from the fuel gauge sender unit and processes it for display. This is not like an old fuel gauge on a Triumph Spitfire.
I believe the level sender is either obstructed or damaged, or possibly something has been over tightened. The fault lies at the door of Halfords, you need to remind them that their work needs to be of 'a reasonable quality' and give them them an opportunity to 'rectify the fault in a timely manner'. The Sale of Goods Act should apply here.
Driving 300 miles on a full tank will leave you enough margin to refill safely (assuming you're no boy racer) I always drive to distance rather than the gauge, as I'm not that keen on it's non-linearity. It hardly moves on the first 100, drops to 1/2 at 200, then heads toward E at 300, I fill up at around 350 miles.
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TG
* The dirt is not mine, this is a scrapyard car.
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