If you are going from a LAN device to a LAN device, there should be nothing passing over the internet in your setup.
Even CAT5 will do gigabit over a 100ft cable all day, and you only have a gigabit network adapter in that computer (and probably any other device in in your house) anyway. There isn’t a huge cost difference between CAT5/CAT6, and for a 1G network with shorter cables like yours there isn’t really a performance difference, but I hope they sent you an honest cable. CAT6 can do 10G for 55 meter. The cable isn’t your limiting factor unless it is damaged (this doesn’t seem likely based on your description).
File transfers are measured in MB/s (megaBYTES per second) and network speeds are typically measured in Mb/s (megaBITS per second). There are 8 bits to a byte, so you have to convert. In a perfect world, with gigabit adapters and a cable capable of full throughput, you can get a theoretical 125 MB/s. This is the maximum speed, but it does not account for overhead or other activity.
You also have to factor in the limitations of the disks. A NAS with 2 drives in a RAID1 (mirror) will only be as fast as 1 of the members. Most drives max out around 150MB/s, but some are much slower (typical “NAS” drives on the cheap end are 5400rpm which are slower than the 7200rpm drives typically found in desktop computers. Laptops often have 5400rpm drives as well). Sequential data is much faster than random data, which is what you probably have unless the drive was defragged shortly before you started the transfer (don’t do this, you’ll spend more time than just waiting and it’ll tax your drive). 20-50MB/s is pretty good normally for a single disk with normal data.
Another factor is the type of data being moved. 1x 1GB file will transfer faster than 1000x 1MB files. If you have lots of smaller files and a lot of data the initial sync will take a LONG time, but in the future you’ll only have a few files to sync at a time and it’ll go faster. Watch the file transfer and you’ll probably notice that the speed increases when it hits a larger file and slows down when there are lots of small files.
Do you have other devices putting data on the NAS at the same time? This will slow the process down.
If your ISP provided router is not great (most aren’t) they can have a hard time with large data streams (research “bufferbloat” for info on how that works). Essentially they choke when you put a large load on them. It also may only have 100M ports, which could slow you down. Do you have a network switch in between the router and the devices? If yes, check and see if it is only “10/100” or 100M instead of “10/100/1000” or 1000M/1G.
Another possibility is that you are actually transferring things over Wi-Fi (both can be connected simultaneously). Try disconnecting from Wi-Fi before transferring the files and see if it is faster.
Lastly, 250GB is going to be a pretty long wait unless you have that workload distributed over more disks or faster disks and use a faster link (10G is pretty cool).
If you are just moving files with file explorer in Windows, I’d recommend doing it in smaller batches so if you have an issue in the middle of a transfer you don’t have to start all the way over again.
EDIT: after typing all of that I saw that you are connected at 100M, so that is the reason for the slowness… Check any device you have in between that computer and the router to make sure it can do gigabit. If there is nothing, it may be the router only has 100M ports.
Take a picture of your router or provide the model info so we can research.