Hi Annie, if you are seeing an ENT consultant (as opposed to a gastroenterologist) then probably your reflux is what's called silent reflux, which is when the valve at the top of your oesophagus leaks so that acid and other stomach liquid gets into your windpipe and can even reach your lungs - it's well possible that this could be causing your cough.
Try sleeping on your left hand side with the head of your bed raised, or a wedge pillow - the entrance to the stomach is a little down on the right hand side, so doing this keeps this valve (the LES, lower oesophagul sphincter) higher up which helps gravity keep stomach contents in the stomach. Make sure you leave at least 3 hours after eating before going to bed and work out your food triggers for reflux - there are lots of sites which give this information.
But don't be fobbed off by your GP if you are worried about your cough. It likely is due to the reflux but all possibilities need investigating for persistent cough as MrsM suggests above.