Tricky! As soon as I saw it I thought two things. First, was The Lady Killers. Second instinct was immediately "Peter Sellers".
And I know it's not The Lavender Hill Mob, that was too early to feature an Austin J2, they'd have still been on the basic J-Type at that time.
Well I watched The Lady Killers over Xmas and I remember I was quite disappointed with the lack of classic vans etc, and they used an Austin FX3 taxi to cart the trunck of loot around. So that probably rules that out.
Thinking aloud, I'm guessing it's from a Sellers film? But which one? God knows, it's probably either I'm Alright Jack, Two Way Stretch or The Wrong Arm of the Law. But it's hard really say which, though the blokes in your pic do seem to be on some sort of a blag, so probably not I'm Alright Jack. So one of the others?
I love my old B&W Ealing comedies, just don't ask me about modern films!