Prime minister's trip to Southampton would have taken a little over an hour by train.
Participants in Southampton, Bristol, Cardiff, London, Leicester, Sheffield, Manchester, Dundee and Belfast can expect to be contacted in the second half of this week.
But the health secretary says he is not "putting a date" on the rollout.
The saliva test is being trialled on frontline workers in Southampton.
The headteacher said she made the decision after an “unprecedented level” of staff fell ill.
Redbridge Community School will not re-open until Wednesday.
The film shows the 13-year-old before her body was found in Southampton woodland.