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.

The Ulster county held a commanding 0-12 to 0-6 lead at half-time but the visitors hit back to claim a place in the third round of the qualifiers.
Tipp rattled off seven points after the break to level the match as Cavan managed just one Niall McDermott score.
Conor Sweeney's penalty and a second goal by Robbie Kiely sealed the win.
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"I am gutted. I thought we were fabulous in the first half, playing some of the best football we have played this year," said Cavan manager Mattie McGleenan.
"But you have to play for two halves, and we didn't.
"In the first 15 minutes of the second half we just could not win a ball. Every breaking ball went to Tipperary.
"I felt the referee gave them a couple of soft frees at the start of the second half and they gained a bit of momentum at a critical stage of the game.
"Tipperary were hungry and their movement for the goal was class. We were caught ball-watching."
Gearoid McKiernan, Martin Reilly, Tomas Corr and Caoimhin O'Reilly each scored two points as Cavan built that six-point interval advantage.
But Tipperary started the second period strongly with four straight scores by Conor Sweeney, Kevin O'Halloran, Jack Kenedy and Brian Fox.
McKiernan landed a free for the hosts but it proved a brief interruption to Tipp's scoring spree and they levelled through scores by O'Halloran, Kennedy and Philip Austin.
Sweeney's penalty - awarded for a Jason McLoughlin foul on Austin - put Tipperary three points up, and when centre-back Kiely netted they led by 2-13 to 0-14.
Cavan reduced the deficit with a couple of late points - but the damage had been done.
Isle of Man rider Billy Redmayne has suffered serious injuries in a crash at the Oliver's Mount Spring Cup meeting in England.
[NEXT_CONCEPT]
Cavan's All-Ireland Championship campaign came to a stunning end as Tipperary staged a remarkable comeback at Kingspan Breffni.