The school alerts followed a series of bomb hoaxes last week that hit Italian airports and airlines elsewhere in Europe, prompting scares, evacuations and widespread disruption.
In Northern Ireland, education officials told schools to remain open unless otherwise advised by police. Some local authorities confirmed that some schools had been evacuated.
Police were treating a threatening email sent to more than 300 schools as a hoax. Scottish police said the warnings were"immediately assessed"as hoaxes.
Italian officials said a single email appeared to have been sent on Wednesday to the inboxes of 6,300 of 7,476 state schools. Private schools were unaffected. More than 6,000 Italian schools received a threat.
Police Scotland said there was no requirement to evacuate schools or keep children away from education based solely on the content of the emails.
Scotland's education secretary, Mairi McAllan, said there was no risk to the public. She said specialist cyber-crime officers were working to identify the source of the messages.
"As a parent and as a constituency MSP I understand just how concerning this will be for parents, for pupils, for carers and for school staff,"McAllan said.
Although many schools were initially evacuated on Wednesday, the alarm was quickly lifted and pupils were allowed to return to class.
"The text seemed almost comical,"Barrella said."It was like a message from Martians: 'We have come to annihilate you.' I even had a bit of a laugh.".
Some parents in Milan decided to take their children home for the day. The majority of schools in the city did not send their pupils home.
Some 280 school head teachers had already received threats in recent days, according to Italian reports.
A Russian extremist group denied sending the hoax bomb threats.
Although a number of threats were claimed by an extremist misogynistic group, it later denied any link to the false warnings.