Answering conjunctive queries (CQs) over EL knowledge bases (KBs) with complex role inclusions is PSPACE-hard and in PSPACE in certain cases; however, if complex role inclusions are restricted to role transitivity, a tight upper complexity bound has so far been unknown. Furthermore, the existing algorithms cannot handle reflexive roles, and they are not practicable. Finally, the problem is tractable for acyclic CQs and ELH, and NP-complete for unrestricted CQs and ELHO KBs. In this paper we complete the complexity landscape of CQ answering for several important cases. In particular, we present a practicable NP algorithm for answering CQs over ELHOs KBs—a logic containing all of OWL 2 EL, but with complex role inclusions restricted to role transitivity. Our preliminary evaluation suggests that the algorithm can be suitable for practical use. Moreover, we show that, even for a restricted class of so-called arborescent acyclic queries, CQ answering over EL KBs becomes NP-hard in the presence of either transitive or reflexive roles. Finally, we show that answering arborescent CQs over ELHO KBs is tractable, whereas answering acyclic CQs is NP-hard.