I have experimented with several different boot scooters, including going up to a Rascal Vista DX to get the extended range (supposedly up to 34 miles) but as I am a fairly big guy at 6′ 3″ tall and 16 stone I don’t get anywhere near stated mileage with any scooter.
There are several other considerations you may wish to consider though, especially as you say your mum has a weak arm/shoulder.
Firstly, leaving a scooter fully assembled creates problems when it comes to fitting it into most car boots and trying to lift the seat off and fold the tiller down can take some doing. I would consider myself very strong when it comes to upper body strength yet still find undoing the locking nut of several different scooter tillers quite stiff to undo. Lifting the seat off can be even harder, especially as some seats are quite wide and heavy. My balance is pretty rubbish so I struggle to lift the seat off most scooters and my wife finds it impossible, despite being able bodied and quite fit.
Leaving a seat on but just folding the back down and the tiller over will still mean that you (they) need a car with a boot height of about 78cm, but that’s not the only problem, as the boot will also need to be fairly square, otherwise the lift cannot lift the scooter in far enough to provide clearance to shut the boot.
Some boot scooters fold up with everything attached, but the two I have tried were not particularly stable and both had weight limits that ruled them out as possibilities for me, but they might work for your dad, however, they would need to be laid flat after folding to fit into a smaller boot and any boot hoist would struggle to reach down to the floor and then be able to lift the scooter up into the boot as the lifting arm doesn’t have the room to move that far.
Having tried multiple scooter types, I have tried and returned, then tried and returned again to Pride Mobility scooters. They seem to be the M&S of boot scooters, don’t cost the earth and have proved much more reliable than most. Resale costs for them on eBay seem to be far higher than most makes too, which is probably because of the reliability. I currently have a Pride GoGo elite traveller sport as my main boot scooter (the sport bit just means that the floor pan is 2 inches longer than the standard traveller) which I need because of my height. I also have two battery boxes, one containing lithium batteries for the range and the other containing lead acid batteries that I use when taking the scooter abroad. Leaving the seat on, with both arm rests in place but the tiller folded forward, I need 78cm height clearance and 94cm width to fit the scooter into the boot. My hoist is an 80kg Autochair Smartlifter.
