can you tell us if your wife was previously married to the father of the kids. and why didn't you give the whereabouts of the father?
i have a friend who has been previously married to the father of the kids, they disclosed everything i.e. the whereabouts of the father and that he is absolutely giving all the rights to my friend and he even said that if the eco would like to confirm the details he gave his phone number and address as well. the father of the kids is living with his new family and the custody was given to my friend. they got their visas and they are now all british citizen.
i am sure you got the right to appeal and if you could prove that the mother is the one providing all the needs of the child (i.e. financially and emotionally) then i think you have a good chance of having the decision overturned. this is what i think but i am sure there is someone in this forum who is more experienced as they have been thru this. i hope you get their visas soon. fingers cross as i am about to file my kids apps as well and looks like i am into a bumpy ride as well.
keep us posted wish you all the luck.