Unfortunately, I think that if you aren't tested for everything at once, there is no way to know for sure...and as it has been said, often there is no way to know which came first.
I had mono when I was 16. I feel that I recovered. Once out of college my health went down hill (actually during college, which is when I believe I was bitten for the first time) When I first saw someone LL, my WB came back neg, but based on my symptoms, ruling out everything else, and my CD57...he concluded I had Lyme. I was not willing to accept that so...fast forward 4 years. My then PC thought I had CFS and tested me for everything under the sun, including all sorts of viruses. I was IgG pos. for HHV-6, EBV, and Mycoplasma Pneumonae.
Twice, I was on Valtrex 1g bid for 6mo. Did it help? Don't know. I know it takes a LONG time, though I am not sure why...I decided to go off because it does stress your liver and I am already on a high dose of doxy. I am hoping to start on some herbal protocols that help with viruses...I agree that both have to be treated but as far as what came first? No idea. I do find it VERY interesting that so many of us ALSO have these viruses...and I too, have Hashimoto's...