Do there exist groups with word problems in arbitrary P-degrees?
I think this is not known. (I apologize - I think I was also one of the people that said I had remembered reading this somewhere.) For example, I believe that Sapir-Birget-Rips, Annals of Math 2002 were the first to show the existence of a group with $\mathsf{NP}$-complete word problem (which would be a trivial consequence of the result asked for in this question). Their Corollary 1.1 states:


  There exists a finitely presented group with NP-complete word problem. Moreover, for every language $L \subseteq A^*$ for some finite alphabet $A$ there exists a finitely presented group $G$ such that the nondeterministic time complexity of $G$ is polynomially equivalent to the nondeterministic time complexity of $L$.


While the second half of this corollary is kind of in a similar spirit to this question, it is a far cry from proving that every $\leq_T^p$-degree contains a word problem.