How does the questioner pay zakat on money saved from British government benefits for her children, and can she give charity or contribute to building mosques in her home country or in Britain?

You must pay zakat on your children's wealth if it reaches the nisab (minimum threshold), and you must pay it for past years, as zakat is not forfeited by the passage of time. This is the view of the majority of scholars, who cite the general evidence for the obligation of zakat and the actions of the Companions as proof.

What you paid to travelers with the intention of zakat, and which they spent on its designated recipients, is sufficient and can be deducted from what you owe. The recipients of zakat are specified in the Quran, and it is not permissible to pay zakat for building mosques or other purposes that are not among its designated recipients. If you are unable to pay zakat in your country due to the lack of eligible recipients, you may appoint someone to pay it on your behalf in another country.

Other charitable acts and sadaqat (voluntary charity) must come from your private wealth. It is not permissible to dispose of your children's wealth except for what benefits them, such as paying the obligatory zakat. There is no specific zakat or sadaqa for Eid al-Adha; rather, what is prescribed on this occasion are the sacrificial offerings (adahi).