I've no real idea on what the Scottish Law says tbh. Can you get a SJIB card? For sure, your grading will depend on your quals. Can go self employed also. It's likely than for other than agency work the SJIB Card is useless.
You certainly don't need a Part P scheme, unless you plan to do work for yourself in domestic premises in England.
You will probably find its similar to Northern Ireland - if you want to do domestic/small commercial work just be insured.
If you want to do Council/Goverment and a lot of Insurance work NICEIC/ECA registered.
Larger Commercial Work/Industrial again NICEIC/ECA registered. You'll probably have to look mainly at the NICEIC route, iirc ECA required a minimum turnover and NICEIC is also a lot more well known. The NICEIC Domestic Installer option is only for Part P as well, so you'd have to be going down the AC route.
There could be little differences in Scotland, I've never worked there but that's it as far as I can tell. The JIB had a chance to regulate the industry but I think they blew it!