Another Scot here, who would always say, DH and I. It's how we were taught grammar at school.
When I was teaching, anyone who wrote or said: Me and Jimmy, was told the correct way to say or write it.
I would even say that in casual conversation and I'm certainly not posh. 
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My dad spent 40 years mastering his craft, and last week someone half his age told him how to eat.😠

