it is rather unlikely that they will delay the flight to TPE except there are more passengers than just you affected and delay is not longer than 45 minutes or so.
They are able to rush you directly to your connecting plane within 15 minutes, luggage inclusive
if you can't make it normally you will be on the second flight to TPE that day. It depends on your negotiation skills and the arguments (infant and toddler), whether they will put you on the earlier EVA flight, i do not think they will put you on another flight as most are by budget airlines, which leave from Don Muang Airport (except China Airlines). Maybe they will provide you lounge access or other special care...
hope everything goes well and let us know of the outcome...