From my own experience.

1) Plunger on shower connector not staying up:

Temporary cure - use a plastic peg with a bit carved out to fit the
spindle to keep it up.

Permanent cure - replace the tap / shower unit with one that has a plunger
that you press down for the shower - Wickes do them.

2) Cold water: even a tank just a few feet above the shower head should
give enough water flow unless there is an obstruction.

Check the cold water feed pipe between the bath cold tap and the tank for
an isolation or gate valve and make sure that it is fully open.

Look in the tank where the cold feed pipe emerges - is anything stuck in
it like loft insulation reducing the feed? I've had that happen and had to
blow it out by sending mains pressure water up the pipe.

3) Hot water:

Is there an isolation or gate valve on the hot water pipe to the bath? If
so partially close it to reduce the flow to be similar to that of the cold
water or fit one.

Even without that, provided the taps are as shown in the link you gave ie
not a ceramic disk but the older type with a washer that screws down
requiring several turns of the tap to turn it off, provided the washer
isn't badly worn, you should be able to turn it on slightly to give a slow
flow of hot water to match that of the cold tap fully open. Then use the
peg to keep the shower working. This will allow a slow shower of the
correct temperature (provided no one else uses a tap or tioilet fed from
the same cold water pipe).

4) Increasing the water flow now the temperature mix is OK:

If the shower hose is an old coiled metal type then the narrow plastic
tube inside is likely to be distorted and reducing the flow - replace it
with a new wider reinforced plastic one.

Descale the shower head or replace it.

If they still wqant a faster flow and don't want the disruption of
replumbing then do what I did fit a Triton T40i booster shower pump. This
is just a pump (not a heater) which only takes 90W maximum so is easy to
fit - I drilled through to the next room and either wired in a new
isolation socket or temporarily just plugged into a socket. The shower
connector from the taps is connected by a hose to the inlet and then the
shower connected to the outlet. With it turned off and the water running
into the bath, turn the cold tap on full, adjust the temperature by
slighty turning on the hot tap, pull up the plunger and fix with the peg,
once some water is coming from the shower head turn on the pump. You will
then get a good flow of water at the right temperature. NB To avoid damage
the pump must be turned off before turning off the taps. - that worked
until we had time for the disruption of plumbing work.

Good luck

Alan

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Using an ARMX6