Long Cycle in Bounded Tree-Width Graphs using DFS and Dynamic Programming
There is a nice trick to reduce $\geq k$ cycle to finding a cycle of length exactly $t$, for $t \leq 2k$ (I first heard of this trick from Daniel Marx). The key observasjon is that contracting an edge of $G$ may not increase the length of the longest cycle in G, and may only decrease the length by a factor at most two. Thus we can search for cycles of length exactly $t$ for $k \leq t \leq 2k$. If no cycle is found, contract an edge and repeat.

This shoukd be quite a bit faster than the treewidth based algorithms, I believe, especially if you use the sieving methods for finding t-cycles (paths). There are very efficient implementations for $k$-path by Bjorklund et al:

Fast Witness Extraction Using a Decision Oracle, with Andreas Björklund and Petteri Kaski 
Proc. 22th Annual European Symposium on Algorithms (ESA 2014), LNCS 8737, 2014, pp. 149-160. 

This should be adaptable to your problem.