If I took out a mortgage when I had a good credit rating, and subsequently defaulted on an unrelated loan which will impact upon my credit rating, can I still switch mortage providers? In other words, will the new provider check my credit rating again?
Yes, they will check your ICB report and take account of any potential black marks (e.g. the defaulted loan).
Each bank/institution has their own credit ratings which they produce from your ICB report, the ICB report is not a credit rating in itself. Just because one institution approved you doesn't mean a new bank will see you as a "safe bet".
As an aside, have you checked your ICB report? Lots of evidence on AAM that the reports often omit a lot of data and the defaulted loan may not appear on yours (yet).
Contact the Icb (www.icb.ie) and see what you're credit rating is like. If everything is clear on that, you could look at doing a switch (as this is what the new lender will be looking at), if anything shows up on the report its prob not worth your while.