Ref: My old dutch. You say that the origin is cockney rhyming slang and you are right. However dutch does not rhyme with dutchess. Cockney rhyming slang usually has two words with the second (and the one that rhymes)not being spoken. Hence the terrible confusion for anyone not in the know! e.g. apples and pears = stairs, whistle and flute = suit, Barner fair = hair. You might say "I went up the apples in me new whistle to comb me barnet".
I think you will find that My old dutch is from Dutch plate = mate.