Ms Phillips, who was travelling with her four-month-old baby and a toddler, said she landed at 1am and did not get her bags until 4am.
One Singapore Airlines flight arriving from Houston in Texas had to go to London Heathrow while another, which came in from Singapore, was forced to land at London Gatwick.
An Etihad Airways flight from Abu Dhabi Zayed International Airport was diverted to Birmingham Airport.
EasyJet, which operates flights from terminal one, said there were “very long queues” for security and disruption to hold baggage processing, meaning passengers could board flights only with cabin luggage.
It said: “Although outside of our control, we would like to apologise for any inconvenience experienced as a result.
“We are doing all we can and working closely with the airport team to minimise the disruption.”
A Virgin Atlantic spokesman said: “Following a site-wide power outage at Manchester Airport, baggage systems are currently not operational.
“Our teams are working closely with the airport authorities and are keeping our customers up to date with the latest information.
“Virgin Atlantic flights are currently departing but may be subject to minor delays.”