is Breton. Although it probably is the 'young deer' word. It's non-existent as a name in Wales until it is reintroduced by the the Normans.
Alun (as it is currently used) must be a form of Alan (sorry). The river name was not used as a given name during the medieval period; Alan was. Modern parents may be naming after the river; but it seems obvious from the distribution of the name that they consider to be a Welsh form of Alan.