In general there are quasi-isometry classes with uncountably many groups, so the must be groups without recursive presentations in that class, and you can have such classes contain groups with solvable word problem.
The Grigorchuk group is an example of a group which has uncountably many groups quasi-isometric to it, but has solvable word problem. In fact the proofs in this answer, or a paper by Anna Erschler: Not residually finite groups of intermediate growth, commensurability and non-geometricity show that there are uncountably many such groups commensurable up to finite kernel. The basic idea is that the groups has many extensions by a finite group. This means you even get commensurable groups up to finite kernel fail to have "word problem rigidity".
In Erschler's paper, she also proves that if you restrict to looking at recursively presented groups then you do get "word problem rigidity" among groups commensurable up to finite kernel. She asks if "work problem ridgity" holds for quasi-isometry classes restricted to recursively presented groups. I couldn't find an answer to that question (and there was no obvious argument I could think of) This suggests your question in this setting could be open.