We study the evaluation of ontology-mediated queries (OMQs) on databases of bounded cliquewidth from the viewpoint of parameterized complexity theory, using as the parameter the size of the OMQ (plus the cliquewidth, in upper complexity bounds, for increased generality). As the ontology language, we use the description logics ALC and ALCI and the guarded two-variable fragment GF2 of first-order logic. Queries are atomic queries (AQs), conjunctive queries (CQs), and unions of CQs. All resulting OMQ problems are fixed-parameter linear (FPL) and we provide a careful analysis of the dependence of the running time on the parameter, exhibiting several interesting effects. For instance, GF2 with AQs requires double exponential running time unless the exponential time hypothesis (ETH) fails, while OMQ evaluation on unrestricted databases is only ExpTime-complete in this setting. For ALCI, in contrast, single exponential running time suffices. Interestingly, this is due to the lower succinctness of ALCI rather than to its lower expressive power.