What would happen if decomposition did not occur

There would be no release of organic nitrogen molecules back to their inorganic state. Dead material would accumulate to choke out livable habitat. 
Plants fix atmospheric carbon in photosynthesis and some species of bacteria fix atmospheric nitrogen so life would continue taking these into the organic cycle but without decomposing fungi & bacteria the cycle would not complete. However the final step of reduction to N2 is only completed by bacteria. In fact bacteria are critical to every step of the nitrogen cycle. 

Fungi also decompose dead plants and animals so are detrital feeders putting nitrogen into the nitrogen cycle but not as N2. Fungi cannot reduce N2O to N2. So fungi make N2O as their denitrification product. True denitrification, like nitrogen fixing, does not occur in eukaryotes only bacteria.