I had the same problem with my Microsoft Designer Mouse (Bluetooth Low Energy) when dual booting two Windows 10 Partitions. I assume this problem is also the same with two separate Windows 10 PCs.
The problem is each time the mouse is paired the MAC address changes, I assume this is due to the new LE Privacy found in the Bluetooth 4.0 standard as described here.
I solved it with a few additional steps to the commonly prescribed 'link key' method:
Pair mouse with Windows Partition A
Pair mouse with Windows Partition B (Partition A now will not work with mouse as the MAC address and link key changes upon second pairing)
Using Windows Partition B, Find MAC address of mouse through (Classic) Control Panel -> Devices and Printers -> Bluetooth Mouse -> Connected Device -> Unique Identifier
3.1 Alternatively, open device manager, find your Bluetooth device under 'Bluetooth', click 'Details' and then Bluetooth Device Address.
Find the link key in the registry by opening regedit & navigating to Computer\HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\BTHPORT\Parameters\Keys\{Bluetooth Adapter MAC Address}\{Mouse MAC Address} and right click -> export (saving the key and all associated values to a file).
4.1 If you see no sub keys here, right click 'Keys', press 'Permissions...' and add 'Administrators' to the user names list (with Read access). Close and reopen regedit to make them visible.
Find the device key in the registry by navigating to Computer\HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\BTHPORT\Parameters\Devices\{Mouse MAC Address} and right click -> export.
Save the two .reg files to somewhere accessible by Partition A (such as a USB drive or cloud storage)
Start Windows Partition A, import the two .reg files created in step 4 & 5 (by double clicking on them). Restart and the mouse should work on both partitions.
7.1 You may have to add 'Administrators' to the 'Keys' key as described in 4.1 on this partition also if you receive an error on import.
(This is slightly modified to the way I performed it but should work all the same... please let us know)
NB: I also had to do the same for my Bluetooth Headset (Jabra 75t) to make it work on two partitions of the same PC. For this, the headset key value was up one level higher in the registry hierarchy and located at Computer\HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\BTHPORT\Parameters\Keys\{Bluetooth Adapter MAC Address}
rather than Computer\HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\BTHPORT\Parameters\Keys\{Bluetooth Adapter MAC Address}\{Bluetooth Headset MAC Address}