SoGA states that they should rectify the problem “in a reasonable time” or refund you (though they don’t have to offer a 100% refund, they can take into account how long you’ve had use of it).
Like so many of these things, “reasonable” isn’t well defined. I’d hazard though that making you wait three weeks before accepting the goods back is unreasonable.
If you want it fixing, you may have better luck dealing directly with the manufacturer; though legally, it’s the seller’s responsibility. They have to make things right with you, any wranglings between the seller and the manufacturer is their problem.
TBH though, I think if it were me I’d be firmly in the position of “I want nothing more to do with this shower” and pushing for my money back.