The Bob Smith Spring Cup meeting at Scarborough was abandoned following the Manx soldier's accident on Sunday.
A spokesman for the event said the 25-year-old was taken to Middlesbrough Hospital where he is "undergoing treatment for his injuries".
He added that he is "breathing on his own and his condition is improving".
Redmayne won the Junior Manx Grand Prix last year and is regarded as one of the brightest prospects in the sport.
His crash came after the feature Spring Cup race had been restarted when Northern Ireland's TT star  suffering a leg injury.
Johnston was also taken to hospital for X-rays following the incident.
The organisers were forced to abandon the meeting in North Yorkshire because of a lack of medical cover at the course following the crashes.
The feature race will now be held at the Cock of the North Road Races in June.
Isle of Man rider Billy Redmayne has suffered serious injuries in a crash at the Oliver's Mount Spring Cup meeting in England.