You shouldn't need to connect to Lynne's router via BT Wi-Fi hotspots on your BT account. You should be able to just connect to it using the wifi password (which on a BT router is usually printed on a little plate on the outside somewhere).
Maybe you mean that Lynne's BT account has been closed? In which case, you will have to use your data allowance.
With Giffgaff, you can view your remaining data allowance here:
https://giffgaff.com/dashboard
Or you can call 43430 from your Giffgaff Mobile and select Option 2, which gives the option to listen to your balance amounts.
If you keep track of it, then you'll know how much you're using ...