Where ever there are joints/overlaps on a roof there is a risk of water entry if they have not been sealed properly,so likely you are right that the flashings around the chimney should be replaced. As of course you will need to get on the roof and repair from the outside I am not sure if you can DIY. Otherwise it would be pretty simple by just getting you hands on some Flash band (self adhesive bitumen tape protected by an aluminium film) and applying it to the chimney (like the attached) You can find this in B&Q, wicks etc. Go for 15cm x 10mt rolls. Make sure you follow application instructions, dry and dust free surface or use a primer and let dry properly, just cut and stick. App pic should look like this.
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