Which router (manufacturer / model) you are using? What kind of storage device do you use (HDD/SSD)?
Where do you have the storage connected?
Have you ever written a large file to the storage device from a PC and see if the same problem occurs? For me it looks like you have a problem with the USB Speed of the Router…
Please try to read/write to your WD via SMB from your computer and check which speed you get there (take a big file and copy it from your computer to the HDD and then back). So you can figure out if the problem ist the Apple TV or your LAN
So the PC also has the performance problem when accessing the hard disk via SMB.
What can also be: do I assume correctly that the connected HDD is used with NTFS as file system? I suspect your router is running a Unix derivative system. This can read and write NTFS with certain drivers, but it is extremely slow. You can test another HDD and format it with ExFat or Fat32 and then test the speed again. Attention, Fat32 only supports files with a maximum size of 4 GB per file.