How do I stop the black sludge coming from our bathroom faucets?
Have the water lab tested for bacterial content at your water source and at the faucets where the sludge is dripping out. E. Coli means your water source is contaminated.

If you're running PEX, black bioslime in certain brands is a known issue.

Flush with Hydrogen Peroxide as Chlorine does absolutely nothing to the bioslime. 

Also, high magnesium content can cause black staining, but not the goo. 

Another thing to check for is bacteria contaminated sediment filters if you have them. If this has happened you will need to remove the filter elements and attempt to clear the systemic infection.

In the case of a biofilm contamination event, it's a multi step process to clear the biofilm. Completely empty the water system, blow out with air, allow to dry to break down the biofilm. Flush with Hydrogen Peroxide then Phosphoric acid. Flush with a detergent to wash the biofilm residue out. Flush with a Hypoclorite solution. Flush with clean water. Test for bacteria.