  We propose an explicit drift-randomised Milstein scheme for both
McKean--Vlasov stochastic differential equations and associated
high-dimensional interacting particle systems with common noise. By using a
drift-randomisation step in space and measure, we establish the scheme's strong
convergence rate of $1$ under reduced regularity assumptions on the drift
coefficient: no classical (Euclidean) derivatives in space or measure
derivatives (e.g., Lions/Fr\'echet) are required. The main result is
established by enriching the concepts of bistability and consistency of
numerical schemes used previously for standard SDE. We introduce certain
Spijker-type norms (and associated Banach spaces) to deal with the interaction
of particles present in the stochastic systems being analysed. A discussion of
the scheme's complexity is provided.
