As Fernando mentions, the answer to the question in your 1st sentence is no, because some such complexes have unsolvable word problem in their fundamental group presentations. 

The answer to your question about replacing homotopies by linear homotopies is yes -- just think through what the Seifert VanKampen theorem says about your 2-complex.  The problem is you don't know what linear homotopies to apply as there's too many of them.  One of the "killer" linear homotopies is one where you introduce cancelling pairs $gg^{-1}$. You don't know how much longer you need to make a word before you "spot" a relation.  That kind of thing.