I'm not sure this needs another answer, but I got hooked on the show, so why not. (spoilers ahead).
Black Hole
We don't know that it's a black hole, even with the image. It kind of looks like a black hole drawing matter from the star, but there's a few problems with the image, and I'll get to that later.
A couple times in the show Elaine Renko (Agyness Deyn) says "I don't know what it is". She's presumably seen more of the data than anyone alive outside of the government insiders, so the show is careful not to commit to a specific scenario and the image only gives a little bit of information, so it could be a number of things. Because they never said what it is, it's harder to criticize the science.
Sudden appearance
Sudden appearances happen. That's not to say something suddenly appears out of thin air, er, space, or quantum improbability or some kind of other dimension nonsense, but "it suddenly appeared" is actually somewhat reasonable because space is big and dark and something going unnoticed until it's right on top of us, is in the realm of possible.
The Chelyabinsk meteor of 2013 is an example of a destructive object just "appearing" in the sky because, as a sun-side meteor it was harder to see and it was smaller than the meteors that NASA tracks, so nobody saw it coming until it was a bright flash in the sky.
Why NASA but nobody else.
NASA has the best telescopes, though there are some very good ones around the globe and numerous amateur astronomers, so that particular aspect may not be probable but it's theoretically possible.
Firstly the show is set in London and everyone seems amazed when the
image below rises from the dusk yet clearly it's been in existence for
the time taken to create the spiral (hours at least?) so nations east
of the UK would have seen it for that time and reported on it globally
surely?
I don't like the spiral. It's pretty but I don't like it. (I'll get to that shortly).
On the suddenness of the visual at the London sunrise, consider what we see. Two cops are chasing their suspect. Other cops are in the garage with their boss who was shot a bit before. Some paramedics are there, taking care of the shot person. The cops chasing grace aren't going to stop and check their phones, so it's entirely possible that images began flooding the internets minutes before and the people in the show weren't aware.
How long was it visible?
You're correct, if it's visible in London at sunrise, it's visible east of London earlier.
Basically everywhere on Earth where you see the sun, you're seeing the same side of the sun. The sun is in different parts of the sky but we all see the same side of the sun, so, unless there's clouds blocking the view, everyone east of London, still should have seen what they saw. That raises the question, how long was the object visibly affecting the sun because it couldn't have been long or word would have spread.
The object appears in that image to be about one solar radii distance from the Sun's surface, assuming the angle is somewhat straight on, not viewed sideways, then we can estimate it's distance from the sun to be about 1 solar radii or 700,000 km. The black dot is also quite small, maybe planet size, certainly smaller than Jupiter (which is 1/10th the diameter of the Sun), it appears larger than Earth (1/100th the diameter of the sun) and that would make it too large to be a black hole or neutron star, but perhaps a white dwarf, super earth, mini gas giant planet or planetary core fragment. There's a fair bit of wiggle room in there.
The solar gravity, basically the escape velocity in reverse would have added about 430 km/s to whatever initial velocity relative to the sun that the object had as it approached the solar-system. Most near-by exo-solar system objects have a relative velocity to the sun in the 1s or 10s of km/s, at least for those objects that orbit in the same direction through the milkyway, but hyper accelerated objects can reach into the 100s of kms per second relative to the Sun. If that was a hyper-accelerated object approaching from the opposite side of the sun than the Earth, it's possible that it would be both difficult to see, and that it approached the solar-system very quickly. The slower something moves, the more time there is to see it and it's gravitational influence, but a hyper-velocity, behind the sun approach might work, either impacting or making a close pass by the sun.
Assuming that object is close to the sun, which it appears to be, even at hyper velocity, to the naked eye, it would appear largely still. If we give this object a relative velocity to the sun of lets say 700-1000 km/s. It would still take, from our perspective, 700-1,000 seconds (12-16 minutes), to travel one solar radii or 1/4 of 1 degree across the sky. That, in theory, means that it might have only been visible for 12-16 minutes, or less if it was traveling faster. That's a short enough time-frame that the cops could have chased their suspect unaware of the images that began flooding the internet. A 12-16 minutes actually works and is theoretically possible, if hugely improbable.
My real question is more around is there any potential scenario where
a black hole of that size could develop suddenly and get to that size
quickly enough for that image to be a surprise? My understanding, very
much as a scientific-layman, is that a black hole of that size would
have to have the mass of at least one of our suns to get to that size
(probably more) - is that a valid assumption?
You're right. Black holes have a minimum mass of about 2 and change solar masses and that much mass simply wouldn't go unnoticed, even with it's tiny size, it's tidal effects on the orbiting planets would be noticed even by armature astronomers "Hey, Neptune isn't where it's supposed to be" - that kind of thing would get noticed. Also, as it passed close to the sun it wouldn't just draw a spiral off the sun but it would make the sun distinctly egg shaped. That much mass wouldn't work, but we don't know the objects mass. We don't know if it's a black hole or what it is.
Black holes don't suddenly develop either. They're made in the cores of very large stars when they go supernova. So your thinking is correct there too.
Where has the mass come from? Also wouldn't the sudden existence of
something else with at least the same mass as our sun suddenly
appearing at the centre of our solar system have enormous
gravitational impact on everything in the system, including Earth?
We'd have massive volcanic/tectonic impact wouldn't we?
The harm from a stellar mass object passing through the solar-system would be that it would change the planet's orbits and changing the Earth's orbit could be bad, pushing us outside our Goldilocks zone, but a 1 or 2 solar mass object would need to get pretty close to the Earth to cause the kind of tidal effects you're talking about. That object appears close to the sun, and we don't know it's mass so the kind of tidal effects you're talking about aren't a problem.
The Spiral
I don't like the spiral because it looks like the object is pulling gas away from the Sun, but it's not changed the shape of the sun. The spiral also appears well developed which would take some time. I don't want to say it's impossible but the spiral around the object, while pretty, seems the least scientific part of the event to me.
Apologies for the long answer, but I wanted to point out that an unnoticed, very fast effect is theoretically possible, just hugely improbable.
