There are naturally occurring groups that have undecidable algorithmic problems. For instance, $F\_2\times F\_2$ has undecidable generalized word problem (membership problem for subgroups) and there is a semidirect product $\Bbb Z^4\rtimes F\_4$ with undecidable conjugacy problem. But to the best of my knowledge every finitely generated group with an undecidable word problem is directly constructed from a Turing machine variant or a non-recursive, recursively enumerable set.



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Of course this problem is not well-defined because `natural' is not well defined in this setting. It should somehow mean things built up from well studied classes of groups (hyperbolic groups, polycyclic groups, solvable groups, linear groups) via natural operations, e.g., extensions, amalgamations, maybe direct limit (I am not sure I like allowing this last option since some strange groups are direct limits of hyperbolic groups). 



Can there exist a `natural' finitely generated group with an undecidable word problem?
If you take a subgroup $H$ of $F\_2\times F\_2$ with undecidable membership problem and take the HNN extension of $F\_2\times F\_2$ where the stable letter commutes with $H$, you get a (finitely presented) group with undecidable word problem. I do not know how natural it is since $H$ encodes a Turing machine. The only known "natural" example of universal algebras with undecidable word problem are free modular lattices (Freese, 1980). 


As a more natural examples you can take McKenzie-Thompson group that simulates all recursive functions or Valiev's (Boone-Collins) universal group that simulates all Turing machines, see comments below. 


