Between 130 and 270 women are thought to have had their lives cut short by the error which was caused by "administrative incompetency".
At least 450,000 women between the ages of 68 and 71 did not receive their notices since 2009.
Some people will receive a letter stating they missed out, knowing they now are terminal ill.
Screenings can help cancers to be caught at an early stage, making them easier to treat.
When the National Breast Screening Service was launched in 2009 the new IT was bolted on to the existing system. Breast Cancer Now CEO Delyth Morgan told the Guardian it was "beyond belief" that the issue continued for a decade.
Public Health Wales will review its breast screening programme "as a precaution" and invite all eligible women for cancer screening.
Screening programmes are run separately across the United Kingdom and so this issue only affects women registered with a GP in England.
He noted that all general practitioners would be briefed on how they should respond, and said the issue was first raised as a potential problem in January.
Breast cancer screening is offered to all women aged 50 to 70 in England every three years; they are sent invitation letters to make an appointment for the test.
Mr Hunt confirmed work is ongoing between the United Kingdom and Welsh governments to ensure the letters are sent out to everyone.
Breast screening in England is run by Public Health England (PHE) and screens around 2 million women a year.
"We encourage women who have been invited for screening to see it as an important step in helping to look after their health, and to therefore continue to attend for screening if invited".
Breast cancer is the most common type of cancer in Britain, with more than 55,000 women diagnosed every year and almost 1,000 dying of the disease every month, according to non-governmental organization Breast Cancer Now.
It intends to contact all 309,000 women believed to still be alive by the end of May.
GPs' leaders said they were "shocked" to learn of the error and said the implications for Global Positioning System would potentially be "significant".
"It's hugely significant, we have to be concerned generally about confidence in the screening process, but we need to know how it has happened".