The use of 'cheers' AND 'Thanks' at the end of an email is, as you suspect, redundant in this context, as the familiar 'cheers' does imply both 'Thanks' & 'bye'.
I stress that 'cheers' is a familiar, so its usage depends on the tone of your email in general.
If you don't know the recipient very well, & are requesting help or information, it's probably best just to end it with the standard British email terminal salutation of
"Thank you in advance,
If I received an email from a stranger, & they used 'cheers' to terminate it, I'd find that to be over-familiar & flippant & slightly disrespectful, as British & US semiology differs quite markedly, as you may have guessed from other posts in here.
Like your husband using 'du' or 'Sie' in any German correspondence. English doesn't have the equivalent familiar tense verbal prefixes, or suffixes, so the familiar tense is suggested by pronouns or little touches like 'cheers', if you know the person well, or they're family.
Depends how familiar you want to be, really, doesn't it?
I may be bonkers but at least I'm British.