Is it natural to describe a common cold as long? For example:
I had a long cold for a week, but thankfully it went away.
Would it be more natural to use the adjective lengthy or prolonged for illnesses?
No, because you already described the length of the cold there is no need to use further adjectives about the length.
Many prefer to use an adjective instead that describes how the cold was. For example,
I had a horrible cold for a week, but thankfully it went away.
Now, if you didn't have "for a week" in your sentence, you could use a similar adjective for the cold's length.
Example,
I had a prolonged/persistent cold, but thankfully it went away.