A good match is to use selective color controls rather than only global saturation/contrast.

Based on the example and community feedback, Nikon’s Miniature Effect appears to combine:

• increased overall contrast
• increased saturation
• a warm yellow/amber shift
• possibly a slight green white-balance tint
• stronger boosting of orange/yellow tones than other colors

In post, try this workflow:

1. Raise contrast moderately.
2. Increase vibrance or saturation a bit.
3. Warm the white balance slightly toward yellow/amber.
4. If needed, add a small tint toward green.
5. Use an HSL/HSB/HSV tool to selectively adjust color bands, especially orange and yellow. Increase their saturation and possibly luminance more than the other channels.

Most editors split colors into bands such as red, orange, yellow, green, aqua, blue, purple, and magenta. Fine-tuning those individual bands is the key step for matching the look more closely than global sliders alone.

So the short answer: use contrast + saturation, then selectively boost orange/yellow in HSL and warm the white balance slightly.