Is there a fuse for the wipers? If so, any idea where it is located?
If you don't have a a service and/or owner's manual, you NEED one! They're for sale on ebay & Amazon all the time, both printed originals/reprints and CDs.
There are two (2) glass fuses on the firewall. Sometimes just surface oxidation is enough to cause high resistance and create problems for our electrical systems, so checking, cleaning (small stainless wire toothbrush) and tightening connections will solve or prevent lots of problems.
Owning, or at least happily enjoying, a Healey is a hands on job; if you can't handle it yourself, then you'd better find a competent mechanic you can trust.