New modem, same time machine, same mac book pro

I can see my time machine on the wif bar but eventhough it "starts" backing up, it never does.

It is not recognizing my computer.

This happened after buying a new modem.

I have been told to reset it.

If I reset it, would I loose all the data in my time machine?

I need a specific file that I deleted from my computer as I was trying to make space so the file vault would not lock me out.

Bought a new external hard drive and it is not compatible with apple.

Can you please help?