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At least you've not got a leak on the supply side, that can be a lot more damaging. I would start with the sealant. I've fixed a few shower leaks that way. Sometimes the coloured (decorative) sealant is not backed up by anything more substantial underneath. If you have a multi tool there is a special blade for cutting out sealant. You might notice as you're removing it that there is some water behind it or signs of water having penetrated. If it's grout then look for the places that dry slowest or just grind the grout out (again with a multi tool and appropriate blade) and re-grout.
New shower is a touch overkill but there are some leaks that defy logic and persist despite your best efforts so you might need to be patient. Do check the trap too if possible.
Yeah it’s wooden unfortunately. I’ve found a couple of guides for locating the leak (probably cracked grout apparently) so I’ll try that first - we’re talking a few drops over the course of a 15 minute shower and I can rip the thing out but we can’t afford to replace it at the moment!