In still air every boundary layer starts laminar. How soon it transitions to a turbulent boundary layer depends on:
- the local Reynolds number,
- the pressure gradient,
- wing sweep and
- disturbances like bugs, rivet heads or turbulators.
Flat plate flow (without pressure changes) normally transitions at a Reynolds number of 400,000. If the flow is accelerated, all speeds in flow direction increase while cross flow will not be affected, so a laminar boundary layer in accelerating flow is stabilized. On modern gliders the lower surface is laminar in excess of 80% chord at higher angles of attack, which corresponds to a Reynolds number of 1,500,000 or more when transition eventually occurs.
On the other hand, a pressure rise in flow direction corresponds to a deceleration in flow direction, so any movements perpendicular to the flow direction will grow relatively to the flow speed, and as a consequence the turbulent transition occurs rather quickly. Upper side flow past the suction peak near the leading edge is a prime candidate for transition, and that is what caused flow around the "traditional airfoil" to become turbulent earlier. The graph in your question is misleading because the lower side flow of the traditional airfoil should be as laminar as that of the P-51 airfoil if the surface smoothness of both is comparable.
Also, with the flight speed of the P-51 very little laminar flow was left; the full effect of laminar airfoils can only be exploited at Reynolds numbers below 5,000,000. See this article for details. The "rooftop" distribution of the 6-digit NACA airfoils did help, though, because it would give them a higher critical Mach number than the "peaky" distributions of earlier airfoils. The suction peak near the nose would lead to local supersonic flow at a lower flight Mach number, and increased drag from the shocks which would follow. Most important, however, was the very smooth wing surface with no gaps ahead of the spar.
Wing sweep will also make it hard to maintain laminar flow. As you know, on a swept wing only the speed component perpendicular to the wing will be affected by it, so the accelerating flow past the stagnation point will curve inwards on a sweptback wing. At the same time, viscosity will slow down the flow near the wing skin. The consequence is a twist in the speed distribution over the boundary layer, which destabilizes the laminar flow and leads to early transition.