Among other uses, the auxiliary verb, would, is used to express an assumption, presumption or expectation in the past. The "would (and in some cases should) + present perfect" is formed with the present tense of have, and the past participle of the verb
would/should + have + past participle
With action or dynamic verbs the progressive form can be used instead to emphasise the activity and to suggest the activity started in the past and was in progress for a period of time.
We saw a police helicopter overhead yesterday morning. | Really? They
would have been looking for those bank robbers.
Despite the appearance of would, this is not the past subjunctive because there is no "if-clause" (protasi) present in the sentence and none is implied. The subjunctive is used to indicate conditions that aren't true, that might have happened in the past, but didn't. In the example above, however, the speaker is speculating as to what the police were doing with the helicopter at that time of the morning. Knowing that a bank had been robbed that same day, he assumes that the police were looking for the bank robbers.
Compare that sentence with this one:
They police would have been looking for those bank robbers yesterday morning (but they didn't), if they had bought a helicopter (but they hadn't bought one).
In the if-clause the police didn't buy a helicopter therefore, they couldn't search for the robbers yesterday morning. The subjunctive mood tells us the hypothetical result of a contrary fact in the past.
Returning to the OP's passage
At home I locked myself in. Jason, my lawyer and Lyn sent texts
simultaneously at the point the announcement was made. Lyn would have
been sending texts consecutively for 15 minutes. Apparently 35 newspapers in the world carried the news on the front page, including the New York Times.
Here the narrator, Sir Alex Ferguson, is assuming what Jason and Lyn were doing while he was announcing his dramatic and unexpected retirement as football manager of Manchester United to his players in the dressing room. During those 15 minutes, Alex Ferguson presumes that Lyn was sending texts to newspaper editors all around the world.
Sir Alex Ferguson, a former Scottish football player and one of the most successful British and European football managers of all time. He managed Manchester United from 1986 to 2013, winning a grand total of 13 Premier League and two UEFA Champion league titles.
Sources: Sir Alex Ferguson My Autobiography
Advanced Grammar in Use Martin Hewings (units 7, 8, 99)
Advanced Language Practice Michael Vince (Unit 12 Modal auxiliaries 2: past, page 66)
A Practical English Grammar A.J. Thomson A.V. Martinet (160 will and should for assumptions; 231 C should/would have expected + infinitive construction)