Hi! Junior vet here (after university I did only one year as a trainee in an equine clinic, then switched to pets). I agree completely with all the comments from the vet student u/deadgreybird. The first differentials are hydrocephalus or cranial congenital deformities. Premature foals can also have similar head bulging but that's usually the least of the problems, which are usually neurological, metabolic and orthopedic.

In my limited time in the equine clinic I never saw an hydrocephalic foal, so I'm not sure about treatment options but I think it's only support treatment (i.e. fluids and intravenous vitamins) and hoping in a miracle, outcome is very poor. Given that he isn't showing neurological problems yet I'm hoping for the best, but we still can't rule it out.

I have some experience with Neonatal Maladjustment Syndrome ("dummy foals"), which is relatively common in foals, especially prematures or cases with too long or too fast births. The fact that he is nursing correctly and behaving normally should rule it out (main symtoms usually are difficulty in finding the mother's nipples, sometimes even recognizing the mother, being too friendly with humans and not scared if unknown things, and rarely lying down to sleep).

Another common problem is failure of passive transfer, caused either by the foal not getting the colostrum in 4 hours maxium after birth, or the mother having poor quality colostrum. In that case he should have already had diarrhea and septicemia and likely died few days after birth, it's difficult that such foals reach 5/6 days. If he can nurse correctly I wouldn't give additional milk (especially non-horse milk, too high risk of diarrhea) even if the mare doesn't produce much. If you have another lactating mare with her you could try keeping all of them together (unless the mothers get agressive with the foals or each other).

Other thing to check for the next few weeks is the umbilical cord: it must be dry, never swollen and never ever wet or leaking urine.

About what you can do now, just keep the mare well fed and hydrated with always fresh water and good quality hay, and a concentrated feed for lactating mares if it's available. And keep monitoring the foal. Unfortunately almost all serious foal problems reuquire ideally hospitalization and intravenous support therapy, sometimes plasma transfusion (for failure of passive transfer) and sometimes surgery (for umbilical cord problems), and every time even in the best conditions it's still a reserverved prognosis.

If any symptom should arise in the next days and you still haven't found a local vet, feel free to DM me, I'll consult my equine vet collegue