The action to cancel Thursday's races was taken with unanimous support of the BHA's industry veterinary committee and will affect meetings at Huntingdon, Doncaster, Ffos Las and Chelmsford.
The trainer concerned is believed to be Donald McCain, whose Bankhouse Stables in Cheshire is home to more than 100 horses and is the only racing stable to send runners to both of Wednesday's National Hunt meetings.
As a result all meetings on Thursday have been abandoned.
Irish trainer Gordon Elliott ran a number of horses at Ayr on Wednesday but his runners were still travelling back home when the news of the positive tests broke and they were quickly placed in isolation away from his stables.
Nearly 300 British-trained horses ran at last year's Festival and 82, almost one-third, made their final start between 20 and 33 days before the meeting.
In a statement, the BHA said: "This precautionary approach is meant to ensure we put the health of the horse population and control of the virus first, and avoid any unnecessary risk that might come from returning to racing too quickly".
"The fact that the cases have been identified in vaccinated horses presents a cause for significant concern over welfare and the potential spread of the disease and the action to cancel racing has been viewed as necessary in order to restrict, as far as possible, the risk of further spread of the disease", the BHA said in a statement.
Symptoms include coughing, a high-fever, and nasal discharge.
The BHA said the full extent of the problem was not yet known but it has worked quickly to identify which yards could have potentially been exposed.
The governing body's statement added: "The BHA is presently communicating with yards potentially exposed to ensure appropriate quarantine and biosecurity measures are put in place and horse movements restricted to avoid possible further spread of the disease".
Why is horse racing cancelled today in UK?
Transmission of the disease can occur through indirect contact, for example through people, and it can also be airborne over a reasonable distance.