Using lambda calculus to derive time complexity?
There are quantitative results concerning $\lambda $-calculus, in the form of measuring the length of reductions in (typed-)lambda calculus. But this is of course far from saying anything about the complexity of algorithms (especially that the bounds obtained are fast-growing). See for example:  Arnold Beckmann, Exact bounds for lengths of reductions in typed $\lambda $-calculus, Journal of Symbolic Logic 2001, 66(3): 1277-1285.

For something closer to your question, there is a current project that develops and studies a type-system (a functional programing language) which by static-analysis can determine (polynomial) run-time bounds of programs (as well as other resources used by programs). So in some sense, this might hint that there might be some advantage in using functional-programing for analyzing  run-time complexity. The project homepage is here.

A possibly representative paper of this project is: 
Jan Hoffmann, Martin Hofmann. Amortized Resource Analysis with Polynomial Potential - A Static Inference of Polynomial Bounds for Functional Programs.  In Proceedings of the 19th European Symposium on Programming (ESOP'10).link