Thanks for your question.
If you are to return to Italy shortly on the basis of the Italian permit that you have then the best thing to do would be to apply for a family permit whilst you are back there.
This is permit for a non-eea national spouse of an eea national who will work outside their home country. Once you have your family permit you can return to the UK. Once in the UK you will have to apply for a residence card, which is an endorsement in your passport confirming your right to reside in the UK for 5 years.
At the end of the 5 year period you and the children can apply for Permanent Residence of the UK (if you wish) confirming your right to remain in the UK free from immigration time restrictions, so you would no longer require a visa in the UK.
All the information on the above is contained on the following links:-
You should consider seeing a UK immigration solicitor before you leave to prepare the application so that you can submit it to the UK embassy in Italy as soon as you arrive in Italy before chistmas.
If this has been useful please kindly click accept so that I may be rewarded for my time. If you do not click accept your money stays with the site and I do not receive any credit for the time I have taken to answer your question.
I will answer your follow up questions you may have.