  In the synthetic geometric setting introduced by Kunzinger and S\"amann, we
present an analogue of Toponogov's Globalisation Theorem which applies to
Lorentzian length spaces with lower (timelike) curvature bounds. Our approach
utilises a "cat's cradle" construction akin to that which appears in several
proofs in the metric setting. On the road to our main result, we also provide a
lemma regarding the subdivision of triangles in spaces with a local lower
curvature bound and a synthetic Lorentzian version of the Lebesgue Number
Lemma. Several properties of time functions and the null distance on globally
hyperbolic Lorentzian length spaces are also highlighted. We conclude by
presenting several applications of our results, including versions of the
Bonnet--Myers Theorem and Splitting Theorem for Lorentzian length spaces with
local lower curvature bounds, as well as discussion of stability of curvature
bounds under Gromov--Hausdorff convergence.
