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.
In the 1980s she starred with Phil Daniels and Jonathan Pryce in the film Breaking Glass, with her debut album its accompanying soundtrack.
An anniversary tour marking 40 years since its release had been rescheduled to take part in March after being postponed from 2021 by the coronavirus pandemic.
Ticket-holders are being asked to await a further announcement.
Scheduled dates for a joint tour with Toyah Wilcox called Electric Ladies of the 80s remain unchanged, said her management.
"We will keep Hazel's health as the only deciding factor on these dates," said an announcement on her website.
"Her recovery is going to take a while but she is tough and is responding to stimuli and to treatment," said her brother.
"She's going to need patience from us and from herself, I've already learned that she's started to show her usual feistiness," he added.