Royal Navy designations before NATO-style abbreviations

Did the Royal Navy use designations comparable to CVN, CVL, or LPD?

Yes in functional terms, though they were not shortened into the later NATO-style letter codes.

The Royal Navy did use role designations for its ships. What it lacked was the later practice of compressing those descriptions into standardized NATO-style abbreviations. Codes such as CVN or LPD therefore should not be mistaken for the existence of classification itself. Functional categories can exist without the shorthand later used to display them.