This could be a comment, but is probably too long for one.
What does the OP mean by:
The entire planet, as well as all other bodies in the solar system are sent forward in time fifty years, so it's not possible to notice purely from observing bodies within the solar system (in effect, the rest of the observable universe has changed, but not our solar system)
"All other bodies in the solar system" include the Sun, seven other planets, at least 5 dwarf planets, possibly the hypothetical Planet Nine, about a million known minor planets or asteroids, hundreds of moons of the planets, Thousands of Trans Neptunian objects in the Kuiper Belt, hundreds or thousands of known comets, and the million, billions, or trillions of undiscovered comets believed to be in the Oort Cloud, etc.
If any of the known objects are left behind, sooner or later some astronomer is going to take a look at them and discover that they are not where they used to be. And if astronomers keep looking to see if other know objects are are actually in their predicted positions, they will soon compile a list of which ones are and which ones aren't. And by compiling lists of the latest dated observations of various now missing objects, they will be able to get a good idea for when those objects disappeared.
And it seems to me that the angles between the Sun, the Earth, and another body orbiting the Sun are constantly changing as both the Earth and the other object orbit the Sun with different speeds. The period between successive moments when another body orbiting the Sun happens to be at the same Angle relative to the Sun and the Earth is call its synodic period.
Here is a link to a table of the synodic periods of the other seven planets, and 9 other bodies that orbit the Sun, relative to the Earth.
Each object has a different synodic period. So try to calculate how much time will pass before an even number of each and every one of those 16 synodic periods passes and all 16 bodies are at the same angles relative to the Sun and Earth as they are this very moment.
Furthermore, a planet will not be in the same direction relative to the background stars after a full synodic period has passed. For each body, there must be a certain number off full synodic periods which will pass before the planet is both in the same angle relative to the Sun and Earth and also in the direction relative to the background stars. So since those other periods for each body are several - probably very many - synodic periods long, The time that it would take for all those 16 orbits and synodic periods to realign and be the same again should be extremely long.
it is my belief that even though the apparent directions to the stars shift very slowly due to their immense distances, the time for such a realignment would be so long that the stars would have shifted their apparent positions so much that the shapes of the constellations would clearly be different. So I believe that the apparent positions at any one moment in time of those 16 bodies listed will never be the same again in the billions of years of future history of the solar system, that it is a unique configuration.
And there are are many other solar system objects beyond the 16 in that list, which are also routinely observed by both amateur and professional astronomers.
And what about bodies that orbit other solar system bodies?
Amateur astronomers often observe the four Galilean moons of Jupiter, perhaps often enough for them to be under constant observation, which might mean that someone would actually see them disappear in real time if they are left behind in the time jump.
A few decades after the Galilean moons were discovered, tables predicting their movements were published. If someone observed their apparent positions, passing into the shadow of Jupiter or other moons, casting shadows on Jupiter, occulting other moons, passing behind Jupiter and emerging from behind Jupiter, etc., etc., and noted the local time using an accurate clock, they could compare those local times with the times predicted for some other location in the published tables. And they they could calculate the difference in local astronomical time between the place where the observations were made and the place where the predictions were made for. And thus they could calculate the difference in longitude between the two places.
This method was first used to find the longitude of places more than 350 years ago.
And there are tens of other moons in the solar system with well predicted orbits. Those other moons will also have to be moved by the super science and technology, or extremely powerful magic, used in the story, in order to keep the Earth humans ignorant of the time jump for as long as possible.
So if the beings (?) who move the solar system forward in time by 50 years move only ten solar system objects and leave the rest behind, it is quite possible that some amateur astronomer will be looking at one of the objects left behind, and will actually see it disappear from view, and/or be making a video recording of it, at the moment it disappears. So moving only ten solar system objects forward in time may result in instant discovery that something strange has happened.
If the one hundred solar systems objects most likely to be observed are moved forward in time it should take longer to discover what has happened.
If the one thousand solar systems objects most likely to be observed are moved forward in time it should take a longer time to discover what has happened.
If the ten thousand solar systems objects most likely to be observed are moved forward in time it should take much longer to discover what has happened.
If the one hundred thousand solar systems objects most likely to be observed are moved forward in time it should take even longer to discover what has happened.
And so on and so on.
If thousands or millions of solar system objects are moved forward in time and space by fifty years, it would be a very complicated operation, especially making certain that all of them are in their proper positions and orbits relative to each other.
So if the objects are moved in time and space individually, the number of objects which are moved instead of being left behind will determine how quickly professional and/or amateur astronomers discover something very strange has happened.
The best way to delay detection of the move in time and space would be to move everything in the solar system.
So I can imagine some sort of time warp field being generated which covered the entire solar system and moved the entire solar system with many millions, billions, trillions, etc. of objects all at once.
One way in which amateur observers help professional astronomers is by observing variable stars and recording their apparent magnitudes, thus keeping track of the lisght curves of many variable stars.
Another way in which amateur observers help professional astronomers is by observing binary stars and recording the position angles of the stars in them, thus keeping track of the changing apparent positions of the stars and providing data for calculating the orbits of the stars.
And the organizations that the amateur astronomers send their reports to will soon be flooded with reports of variable stars suddenly changing their apparent magnitudes and double stars suddenly jumping around in their orbits.
In the 20th century, astronomers often took long exposure photographs of astronomical objects to show details which were too faint to be seen instantly. And if professional and/or amateur astronomers still do that in 2020, all such photographs of other stars, star clusters, and nebulae taken in time periods including the jump should include duplicate and probably overlapping double exposed images of those distant astronomical objects, as the directions to those objects change over 50 years of time.
The OP wrote:
My first thought is that you may be able to notice the change in position based on parallax of nearby star systems. However, I don't have a great sense of whether fifty years would make a big enough difference to notice with the precision of today's instruments.
Proper motion is the astrometric measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more distant stars.1
Proper motion was suspected by early astronomers (according to Macrobius, AD 400) but a proof was not provided until 1718 by Edmund Halley, who noticed that Sirius, Arcturus and Aldebaran were over half a degree away from the positions charted by the ancient Greek astronomer Hipparchus roughly 1850 years earlier.
So ancient astronomers using naked eye observations suspected proper motion over a period of centuries.
in 1716 Halley proved that Sirius, Arcturius, and Aldebaran had proper motion equalling about 0.162 arc minutes per year, or about 0.972 arc seconds per year.
So in 50 years they might show a proper motion of about 48.3678 arc seconds. Modern astronomical instruments record angles with a precision of 0.001 arc second or better, so they should be able to record the difference in the positions of Arcturus, Sirus, and Aldebaran in a time jump of only one day.
Barnard's star has the largest known proper motion, moving 10.3 arc seconds per year, which is about 0.028 arc seconds per day and about 0.00117 arc seconds per hour, just about detectable at a precision of 0.001 arc seconds.
The Gaia satellite can measure positions of stars with an accuracy of 20 microsarcseconds, or 0.00002 of an arc second, and thus would be barely able to measure the displacement of Barnard's star in a time jump of one minute, and the displacement of a star with typical proper motion in a time jump of a few months.