Personally found the Watford way a bit of a chore and with two kids in tow I wouldn't recommend it. Getting out of the stadium, queuing for the train and the journey itself is a good 90 minutes unless you leave early. Not sure how the contactless debit cards you use like an Oyster card works where kids are involved for buying the train tickets .
As others have said, park on the right side of Wembley (drive or car park), take plenty of food and drinks and even if you end up queuing in traffic the kids will be fed and watered and probably have a kip and you can at least avoid schleping them around North London for an hour and a half.