The process starts with the standard definition (SD) colorized source (in this case, a YouTube screenshot) ..
.. and it can be used as-is. But it really needs to be spiffed up. I use a paint program to get the shot where I want it to go. Once there, Avisynth, or another video program, is used to achieve the same result, even if from different processes in that environment.
First, to get that "fog" of muted color fixed up, I use Color Balance to adjust the ambiance of the shot with a luminance temperature slider. This is moved from it's default position of "Sunlight" (6500K) to the "Incandescent bulb" (2500K) direction. 5600K looked like a good enhancement value -- it expanded the apparent depth of colors without color-washing it. Granted, a subtle effect ..
.. but a step in the right direction.
Next, the colors need some strength. Using the Hue/Saturation/Lightness color wheel, shift the Hue of the entire spectrum slightly towards red while bumping up the Saturation a good amount. However, doing that also over emphasizes an odd splash of aqua on the lapels of the left-hand jacket (and nowhere else). That requires Hue shifting the blue spectrum-segment in the opposite direction while dramatically dropping it's Saturation ... to match the rest of the jacket ..
Now it's starting to look good.
The last and most important correction is luminance for better brightness & contrast. The luminance of the colorized SD source must align to the luminance of the B&W high definition (HD) source (assuming the B&W source is at it's best).
NOTE: unless the luminances are thus synchronized, the final result might have noticeably skewed coloring from the different brightness levels. It's not that big a difference when transferring colorizing between B&W sources. However, using this technique to transfer color between different colored sources will suffer if this alignment is not made.
So, the easiest way to do this is to split out the luminance from both sources (using Split Channel - HSL). Then obtain the setting values by adjusting the SD source to match the HD source (using Histogram - Luminance) ..
SD 360p luminance: start
HD 720p luminance: target
SD 360p luminance: finish
After that, discard those temporary luminance pictures (no longer needed) and apply the obtained settings directly to the colorized source ..
The newly spiffed colorized source now is ready for the H-S-L recombination treatment.