Ask an Immigration Solicitor. Get an Answer ASAP.
With regard to your children this is an example of what the say means penny wise pound foolish. Your children status became invalid after the switch and would be regarded as illegal. You will more likely than not spend more money now regularising thier stay.
When you make your application for ILR your children are unlikely to qualify. This is going to be a mess to untangle.
There is no right sequence of action. You need to apoly for ILR with your wife when she passes exams and both of you qualify. Children will need to apply for a dependent visa
If your children born here did not then apply to be a dependent then the rule you refer to is correct but unfortunately you have regularised thier stay and then didnt apply to include them subsequently which makes them illegal.
What statutory right? There is no such right. Children born in limbo arent illegal but once they have applied for a visa and granted they cannot then claim to be in limbo because they were born previously in limbo. Once they have regularised their status and then allowed thier visa to expire, they become illegal. Being born in limbo means born in the UK while thier parents had valid visas.
Please can you rate? I don't get paid for helping you if you forget to rate
OK please take your time.
Good luck since you are now qualified to give immigration advice. You also said you were seeking second opinion. If you are honest about what you were told you should also disclose that advice here too. The letter means children born after ILR has been granted are entitled to register.
I am happy for you.