From an allergy stand-point, the first goal should be low VOC, which both water-based urethanes and tongue oil (Waterlox) are. There are no such thing any longer as oil-based urethanes or oil based paints, all changed to achieve a low VOC rating by the fed in the last 10 years.

An example here would be that a floor finisher says "I will use the toughest finish I can find" and uses a very strong outgassing 3-4 year old oil-based urethane.

Unless your child has a specific petroleum allergy, often an environmentally triggered type of allergy, either of these floor finishes should be OK, however if he does have an environment allergy such as petroleum based, neither might and you will have to check with an allergist first.

As far as durability goes, I really like either having worked with both. Waterlox is much, much easier to service in the short term, basically, dab some on and buff it a bit, definitely not an option with urethane, however urethane is tougher if applied correctly. The biggest difference between water based and oil based (older) urethanes is the surface flexibility. That is, water urethanes when dry are less flexible at the surface, hence the chips that occur than Waterlox.

If your flooring is soft and likely to realize depressions such as dog claws, furniture, moving wheel tricycle like objects, again, urethane may not be best because the surface is really stretched as the wood below is depressed.

Hope this helps!