In February, Malik announced that his biggest-ever solo tour would include 31 dates worldwide and kick off at Manchester’s AO Arena on 12 May.
The rescheduled tour will now start on 23 May, with just nine dates.
Malik will now perform his first UK show at London 02 Arena on 23 May, and play Manchester’s AO the next day.
Shows in Glasgow, Birmingham and Dublin have been cancelled, but performances in Mexico, Argentina, Chile, Peru and Brazil are still due to go ahead.
Malik fan Morvan Dick told BBC Newsbeat she had tickets for the Glasgow show and was due to attend with her boyfriend.
“We had booked time off work to go, and a hotel,” she said.
“I am actually devastated to be honest, gutted doesn’t really cut it at this point.
“I have been a fan since day one so it’s just a bucket list thing.”
But “health comes first before anything,” she added.
Customers who booked via Ticketmaster have been told that refunds will be issued automatically, and tickets to the Manchester show will remain valid for the new date.
BBC Newsbeat has approached Zayn Malik’s team for further comment.