She was taken to hospital from her home in south-west France on 9 January following a "serious medical event" and spent 24 hours in an induced coma.
Her brother Neil O'Connor posted on social media she had been "showing a lot of progress" since then.
Tour dates scheduled for February to June have been postponed.
The artist, famous for 1980s hits Eighth Day and Will You, played her only 2021 show in her home town of Coventry as part of the UK City of Culture festival.
