Is this string a letter doing a cartwheel?

sed 4.2.2, 30 + 1 -r = 43 31 bytes

Saved 12 bytes thanks to @Neil by shortening the first line

/nu|wm|qb/!d
/^((.).)\1*\2$/!d

Try it online!

Deletes input if falsey, otherwise does nothing to the input.

Explanation

With the -r flag, we do not need to use \( and \) for capturing groups and this saves bytes.

/nu|wm|qb/!                # On every line that does not match this regex
           d               # Delete
/^((.).)\1*\2$/!           # On every line that does not math
 ^((.).)                   #  the first two characters at the beginning of the line
        \1*                #  repeated
           \2$             #  with the first character at the end of the line
                d          # Delete

JavaScript (ES6), 82 78 77 bytes

Saved 1 byte by using two falsy values, as suggested by ThePirateBay and MD XF.

([c,...s],S='bqwmun')=>s.reduceRight((r,a)=>r&a==S[S.search(c)^++k&1],k=s>'')

Test cases

let f =

([c,...s],S='bqwmun')=>s.reduceRight((r,a)=>r&a==S[S.search(c)^++k&1],k=s>'')

console.log('Truthy ...')
console.log(f("nun"         )) // -> truthy
console.log(f("nunun"       )) // -> truthy
console.log(f("wmw"         )) // -> truthy
console.log(f("bqbqbqbqbqb" )) // -> truthy
console.log(f("ununununu"   )) // -> truthy

console.log('Falsy ...')
console.log(f("nunununu"    )) // -> falsy
console.log(f("wmwun"       )) // -> falsy
console.log(f("v^v^v"       )) // -> falsy
console.log(f("AVAVA"       )) // -> falsy
console.log(f("OOO"         )) // -> falsy
console.log(f("nunwmwnun"   )) // -> falsy
console.log(f("nun unun"    )) // -> falsy
console.log(f("nunwmw"      )) // -> falsy
console.log(f("nnuunnuunnuu")) // -> falsy
console.log(f("m"           )) // -> falsy

Python 3, 111 bytes

-2 bytes thanks to Mr. Xcoder.

lambda s:s[0]==s[-1]and any(any({*s[::2]}=={i[j]}and{*s[1::2]}=={i[j<1]}for j in[0,1])for i in['nu','mw','bq'])

Try it online!