While there is no definite answer, we can make some inferences from the text. Based on suggestions in the chapter we can set some upper bounds for particular events.
Firstly, the messengers from Halifirien had already set out by the evening of the 7th of March. This is clear as on the 9th of March Pippin and Gandalf reach Minas Tirith (Appendix B, Tale of Years), and after arriving in Minas Tirith Pippin says to Beregond:
‘When?’ said Pippin. ‘Have you a guess? For I saw the beacons two nights ago and the errand-riders; and Gandalf said that it was a sign that war had begun.’
Return of the King: Book V, Chapter 1: Minas Tirith
In the conversation, Beregond seems to suggest that the beacons were only lit on the evening of the 7th of March:
‘But why were the beacons lit two nights ago?’ [Pippin]
‘It is over-late to send for aid when you are already besieged,’ answered Beregond. ‘But I do not know the counsel of the Lord and his captains. [...]
But if you would know what I think set the beacons ablaze, it was the news that came that eve out of Lebennin.’
ibid.
It is almost certain from Beregond's suggestion that the beacons had been lit sometime that evening (7th March). As such that sets an upper bound of the time between the evening of the 7th and the night of the 7th. Anywhere between 1 and 6+ hours.
So now we have an estimate for how long it took the Beacons to be lit and the riders to set out we can work backwards to find out how long it took Pippin and Gandalf from Isengard to meet the riders, and how long the riders may have had to get to Edoras.
From the "Tale of Years" and "Chapter 3: The Muster of Rohan" we can deduce that it took until the 9th of March for the errand-rider to reach Dunharrow, where Théoden was.
March 8 Aragorn takes the ‘Paths of the Dead’ at daybreak; he reaches Erech at midnight.
Appendix B: Tale of Years
‘I do not know,’ she answered. ‘He came at night, and rode away yestermorn, ere the Sun had climbed over the mountain-tops. He is gone.’
Return of the King: Book V, Chapter 3: The Muster of Rohan
‘A man is here, lord,’ he said, ‘an errand-rider of Gondor. He wishes to come before you at once.’ [...] ‘The Red Arrow has not been seen in the Mark in all my years! Has it indeed come to that?’ [...] ‘But I myself am new-come from battle and long journey, and I will now go to rest.’
ibid.
As such we can deduce that on the night of the 9th, the errand-rider had reached Théoden, and on the night of the 7th he had set out. On the evening of the 7th the Beacons had been lit. Therefore, we can guess that it took somewhere between 1 and 6 hours for the beacons to run their course, and around or just under 48 hours for the errand riders to reach Théoden and Dunharrow. It is possible this process may have taken longer than necessary as the errand-rider may first have gone to Edoras then been redirected to Dunharrow.
In conclusion, from the lighting of the beacons to Théoden receiving the message, there would've been about 48 hours of time, the majority of which was taken up by the errand-rider riding out from Halifirien to Rohan.
As for the number of beacons there were 7, in order from East to West: Amon Dîn, Eilenach, Nardol, Erelas, Min-Rimmon, Calenhad and Halifirien. (Book V, Chapter 1: Minas Tirith)
