Acrylic paint on wood: how to make it "glow" after the painting is already done
I think it's hard to judge the real difference between the paintings your nephew made and the example of the goal he has in mind: the first are taken in what seems a yellowish light, while the second is colour corrected (at least, it seems to have a proper white balance set) and taken in bright light.
Assuming both the eyes of the giraffe and the fur on the hat of the bird are white in reality, this reproduction comes closer to what the artworks look like:

The visual change this inflicts notwithstanding, there are a few remedies to make the colours in the paintings brighter (assuming no coat has been applied afterwards, since that potentially influences the solutions and makes them dependent on the nature of the coating; if this is the case, let me know, and I can update my answer):
Retouch

It's easiest to retouch or repaint (parts of) the paintings.
The saturation of the current hues may have suffered from dilution or mixing with other hues.
Use pure, unmixed paints, and apply them undiluted on top of the existing colours.
The yellow of the bird, for example, seems to be a desaturated or earthy yellow, or a yellow ochre. If a brighter hue is desired, apply an unmixed pure yellow over it.

Glazing

Alternatively, mix a (semi-)transparent colour using clear, unspoiled water and use it to give depth to the existing colours.
This technique is especially wonderful with oil paints (mixed with oil instead of water, naturally) as it gives very rich colours and depth to paintings, but with acrylics it also works quite well. You can even glaze your acrylic paintings with oil paints as well, for additional control and colour richness. For this to work the paints must have dried completely, so as not to spoil them by mixing with the existing layer.
As an example, you can use a diluted or translucent pure red (e.g. madder lake) to glaze the existing red of the Christmas hat.

Use neon colours

Use the aforementioned neon colours to really saturate some of the colours. The range of neon colours is limited, but you can use them to make some of the existing colours, like the yellow of the bird, really pop.


Other answers point out the role the support plays in the saturation of colours. Wood, especially dried, porous, and/or unprepared, may indeed absorb part of the paint or water in acrylics. Apart from their colours becoming brighter, this is another good reason to use undiluted acrylics, as they will effectively form a plastic layer on top of the support. It then functions similar to acrylic gesso, a property on which some of the solutions offered here are based.