All airlines have become quite strict about baggage weight within the past year, as fuel prices have risen substantially, and many airlines have instituted procedures which makes it impossible or at least difficult for the agent to waive these fees. This is especially true when the check-in is with another airline. My limited experience with BA at EDI is very negative -- the most bureaucratic and pig-headed set of agents in the BA system; so if GLA is anything like that, it further compounds your situation.

I suggest putting some of the weight in carry-on (if BA allows any carry-on out of Scotland these days). Or switch the routing to be via North America, which puts you on the piece system (at least 44kg allowance). A RTW ticket is generally the same price as UK-NZ and has this extra baggage benefit.