Cole can be either seen as a pet form of Nicholas, or apparently as an independent name derived from the Old English nickname Cola. See here, here, here, here.
Since the Latin for Nicholas is Nicolaus (or Nicholaus), the former interpretation gives Colaus. In Italian there is the historical precedent of Nicola Gabrini, a 14th century politician, being known as Cola di Rienzo. Colaus would be declined as Nicolaus and with the same vowel quantities: (Nī)Cŏlāus.
The latter interpretation gives a straight Cola, which doesn't sound too unnatural in Latin because it exists as a suffix, e.g. in agricola (agricultor), but obviously has the problem of sounding like a drink. Cola would be declined as agricola.