How do I cover old baseboard (skirting)?
I think this is a problem easily solved by Quarter round. Actually I think this is why quarter round was invented.

If the existing molding is small enough you could possibly cover it up, but it might look even worse if the original molding is too large because the cover would have to be bigger.

If you want to cover the original molding you could get larger molding, and using a router hollow out part of it so the old molding fits inside.  Then simply nail it on.

Other than that you could always ask the tiler if he could fix the problem (at his own cost of course), because it sounds like he did not do a proper job.  If you have not paid him, consider withholding the payment until the job is done properly.