I would start with a waterproof roller blind that is cheerful. No need to match the blue if you choose a stripe - with blue , white and fawn to go with the existing colours that you can't do anything about. If stripes are not your thing there are some blue based floral blinds where you could tie the scheme in with a blue blind but go full on with retro flowers or pattern that suits all ages using the bathroom.
There are lots of examples if you google blinds to go water proof blue roller blinds.
You could also go with a fawn slat blind as that may last better but I suspect it would look too clinical but you could soften it with contrasting colour baskets (- for example navy blue baskets and navy shower curtain).
Depending on the blind you go for you could , when funds allow, buy a stripe or cheerful shower curtain.
I would also invest in around 3 plastic containers for the ledge in a blue or colour that picks up on your blind choice and tidies the shelf up and makes it easier for all to grab the products they want.
Eventually swopping out the grab rails for chrome ones and putting a chrome towel rail above the radiator for a cheerful towel would help.
I have a disabled friendly bathroom and hate how awful and depressing they can be made to look when they don't need to be.
Not sure about the mat idea as would it be a trip hazard for your mum and end up sopping if you have boys?
Hope I have given you some inspiration