You must have a blockage further down the waste pipe. maybe on a bend?
screwfix are doing a 9m drain rod set for £30 mite be an idea to have a go yourself ( from the drain cover back)
Hi .. these plungers are always have a nut or are threaded in. you may need to get a pair of grips on the plunger and try to turn it. does the knob turn? if so it should twist off and i suspect the nut is under that.
if you cant suss it out give me a call and i will try to talk through it...
first turn the water off to the taps. if you have isolators on the pipes under the bath all the better, if not turn off water from the cylinder and header tank located in the airing cubourd. if the knob has a cover you will see a small allen key,take this out. in any event there will be a nut...
In that case it will be the tap at fault. If you knew the make and model you should be able to get the valve to replace it or simply a new set of mixer taps :)
it sounds like the water pressure is to low to raise the water to the shower head. this can happen when the hot tank is on the same level as the taps. im only guessing you have a hot water cylinder and not a combi boiler. in which case you may need to fit a shower pump