So the bath is draining freely? If you can get the side panel off, maybe you can try and force aforementioned 'small bendy wire' back up there the reverse way?
Blockages up basin soil-pipes don't happen too often - not much else than water goes down it.
Is it 'completely blocked' - as in water level doesn't shift over several hours? If not, let it drain out (over several days if necessary) then pour some undiluted drain cleaner down there. This stuff is dangerous - relatively concentrated suphuric acid - but it eats away at the blockage without touching either sanitaryware or UPVC pipework. But these no point unless you can get the acid close to the blockage - it will merely sit at the top of the water-filled pipe.