With regular human interaction, it is very possible that Blaze and Azurine will become more friendly and receptive to human contact.
However, keep in mind that budgies generally need about two weeks to settle in before taming can really start; budgies who are still afraid from the new environment can often seem friendly out of submission. Of course, this may not be the case with Lemon Lime Twist and Cloudy Sky, but it is possible that they will regress a little bit as they settle in further to their new home.
Be sure to always work with them slowly and at their pace; if at any time they seem uncomfortable, it's important to take a few steps back to let them adjust.
Budgies are not very "cuddly" birds and as a general rule don't like petting or holding. However, if by holding, you mean sitting on a hand, shoulder, or finger, then with careful and patient interaction they definitely will do so
However, the timeframe is different for all budgies; some are more outgoing than others and will take less time to appreciate human company.
As long as you and your son and daughter work with them consistently and calmly, they will gain trust in them over time
Best of luck!