Hello & welcome, I am Dr. B, a licensed veterinarian and I would like to help you with this wee one today.
It sounds like this little lad is quite confused. At this age, fledglings are often hoping out of their nests in an attempt to fly. Often the first attempt at flight is not successful and these wee ones will just loiter in the undergrowth under their nest. Their parents will often know they are there and continue to feed them from their ground position until they are ready to fly off on their own (usually a week or two).
Just to note, the night time disappearance is normal, since these wee ones don't feed at night. So, as you noted it is a labor intensive undertaking during the daylight hours (often they want to eat every ~1-2 hours since they have such small stomachs) but unlike human babies, fledglings do sleep through the night. You didn't note if they were feeding him directly (which would be labor intensive) but if the continue to offer bowl based food first before handfeeding this will help get him eating on his own then this may lessen the intervention needed for him.
Since he did appear in the garden yesterday, it is likely his nest is very close to where he landed. Therefore, it would be ideal if your neighbor would let him stay today and just monitor the situation. While the little lad is asking your neighbor for food, they may wish to consider refraining for a wee bit. That way, his parents will hear him call,hopefully appear (in case they were shy to when everyone was out), and take over care of him.
Otherwise, if there is no sign of them by early evening, then you may want to consider intervention. If the neighbors are not keen to provide care for the next week or so, then they can contact their local rehabilitation center or just take him to the local vet practice. In regards ***** ***** a local rehabilitation facility, you can check Wildlife International (LINK), Wildlife Rehabilitation Information Directory (LINK) Wildlife Rehabbers (LINK), or Wildlife Sanctuaries )http://wildliferehabber.com/rehabber-search) or RSPCA (LINK). Otherwise, your local vets will take him (and either use their contacts or perhaps a keen nurse will take him on). To find your local vet, you can check the RCVS Register (HERE) .
Finally, just in case they do want this wee mission for the next few days, I do just want to make note of what they should be feeding this little guy. If he is requiring handfeeding, then we tend to offer a daily ration of ~20 mealworms and soaked cat food (must be soaked to prevent expansion in their stomach). While doing this, we'd want to have a food plate on offer for him to help him get on with self-feeding. This is often about ½cup worth of diet. Items that we often ofer are a 20% earth worms, 30% soaked food, 30% berries and fruit (ie grapes, watermelon, melon, oranges, etc), 5% veggies (no greens), 1 teaspoon suet (5%). So, you could put this down for him and usually they will go from being very needy to totally swapping to self- feeding.
Overall, for the moment, your neighbor should just let this little guy call and fuss. Hopefully, his parents are near and will take over his care for you. If they don't you can consider continuing care for the next week or so. Or if that is just too much, then consider contacting your local rehab center or vets to get him that wee bit of assistance he needs before he is off on his own.
I hope this information is helpful.
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All the best,
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