NOTE: This is a books answer as there is no indication in the show regarding the question.
First of all, it is by design that GRRM remains ambiguous about distances, time and speed. He just doesn't want people to go around with a stopwatch and a measuring tape and find errors in his story.
When a fan asked how big is Westeros, his answer was:
I have deliberately tried to be vague about such things, so I don't have obsessive fans with rulers measuring distances on the map and
telling me Ned couldn't get from X to Y in the time I say he did.
However, if you really must know, you can figure out the distances
for yourself. The Wall is a hundred leagues long. A league is three
miles. Go from there.
But if you turn up any mistakes in travel times by using that measure,
let it be your secret.
When another fan asked about Chronology of Sam's adventures, he replied:
The reason I am never specific about dates and distances is precisely
so that people won't sit down and do this sort of thing.
My suggestion would be to put away the ruler and the stopwatch, and
just enjoy the story.
So it is by design that you don't see much information about distances or time it takes to cover a certain distance for varying means of transport.
But are there any indications in the Books?
Yes there are. Prince Daemon and Princess Rhaenyra used to race to Dragonstone on their dragons and back to Kingslanding almost every day.
Princess Rhaenyra was a different matter. Daemon spent long hours in
her company, enthralling her with tales of her journeys and battles.
He gave her pearls and silks and books and a jade tiara said once to
have belonged to the Empress of Leng, read poems to her, dined with
her, hawked with her, sailed with her, entertained her by making mock
of the greens at court, the “lickspittles” fawning over Queen Alicent
and her children. He praised her beauty, declaring her to be the
fairest maid in all the Seven Kingdoms. Uncle and niece began to fly
together almost daily, racing Syrax against Caraxes to Dragonstone and
- The Rogue Prince
While there are also numerous other accounts of Dragons flying from Dragonstone to Winterfell, Dragonstone to Kingslanding, Dragonstone to Storm's End, King's Landing to Harrenhal etc this is the only account which has a time factor mentioned with the words "almost daily". So that means, a small trip to Dragonstone and back could be done easily within a day. If we minus 7-9 Hours of sleep needed by both riders, minus further 2-3 Hours for getting ready, getting breakfast, feeding your dragons, minus further 2-3 hours for Lunch and Dinner, that leaves us with 10-13 workable hours. If they could easily take two trips to Dragonstone, that means in worst case scenario, it must have taken them 5-6 Hours for one-way trip. In best case, We can probably assume that it would have taken them less than 1 hour for one-way trip. There are also rumors that the Uncle and Niece took those trips to carry on with, uh certain family affairs, away from the prying eyes of the Court. So it must have taken them a few hours to carry on, uh those activities which makes the case for <1 Hours even stronger.
Now that is only less than a couple hundred Miles from King's Landing to Dragonstone.
Now let's a take a larger example. The travel from Winterfell to the Wall. It is a long journey, as per your own inaccurate map it is supposed to be around 600-700 miles away. Queen Alysanne grew bored one day while her husband was busy talking to his Warden of the North in Winterfell, so she took wing and went to the wall. Given the casual nature of the trip, I assume the Queen did not expect it to take longer than few hours (Else she would have at least notified someone or taken someone along with her. King was unlikely to let her go alone for long time.)
"A queen stayed there for a night." Old Nan had told him the story,
but Maester Luwin had confirmed most of it. "Alysanne, the wife of
King Jaehaerys the Conciliator. He's called the Old King because he
reigned so long, but he was young when he first came to the Iron
Throne. In those days, it was his wont to travel all over the realm.
When he came to Winterfell, he brought his queen, six dragons, and
half his court. The king had matters to discuss with his Warden of
the North, and Alysanne grew bored, so she mounted her dragon
Silverwing and flew north to see the Wall. This village was one of
the places where she stopped. Afterward the smallfolk painted the top
of their holdfast to look like the golden crown she'd worn when she
spent the night among them."**
ASOS - Jon
It is possible that she did notify the King and the trip may have taken longer as she stopped midway on multiple points, after all it's Jon's POV who summarizes the whole event, but we do not know it for sure.
Anyways, judging by Alysanne's trip to Wall, Rhaenyra's race to Dragonstone and Prince Jacaerys' trip to Vale~White Harbor~Winterfell, I'd put the estimate to <10 hours.
It can't be emphasized enough, that this is merely my interpretation and speculation. George doesn't want you to worry about these things.
Why aren't dragons used to spy on approaching armies?
As I said before in comments, that is because of the nature of the beast. Dragons are the ultimate weapon of destruction. They can easily destroy any army they come across so why bother with using them for recon flights? Prince Aemond showed this one-dragon army style warfare in Dance of the Dragons.
Dragons have however been used against Armies north of the Wall.
King Jaehaerys I, grandson of Aegon the Conqueror, successfully unleashed his dragons against the Army of King Beyond the Wall.
His Grace gave the babe a pearl ring off his finger to play with, and
told the twins the story of how their great-great-grandsire and
namesake Jaehaerys the Old King had flown his dragon north to the Wall
to defeat a vast host of wildlings, giants, and wargs. The children
listened attentively. Afterward the king sent them away, pleading
- The Rogue Prince