I also have a sort of mop, also from Lakeland, I use that for the higher tiles as can't reach otherwise, they're probably filthy if looked at too closely!

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I usually use a bleach spray and an old toothbrush. A bit time consuming but it seems to do the trick

I used this method for many,many years
then useless Cillit Bang.
Answer is---Dettol mould and mildew cleaner.
Spray and leave about half hour and shower off.
It truly is worth it's weight in gold.£1.50