Effects on cards that are separated by an "and" happen sequentially, so the melee damage will happen before the radiant damage.
From user phantaskippy (tagged as playtester) on the Sentinels of the Multiverse game forums:
Any use of "and" will happen sequentially. . . . All uses of "and" in situations where timing is relevant are intended to be sequential. It may not be obvious, but it is at least consistent.
This sentiment is repeated here by user Rabit (tagged as moderator and playtester) among the discussions about effect timing:
Do things in the order described.
In reading through forum posts, the consensus follows this interpretation.
To answer the question about why this may be important, that is, you want to increase one damage type for a specific reason, Adam the Admin has this to say over on the forums (emphasis mine):
The damage is intended to be dealt individually. This means if it's increased, both melee and radiant are increased by 1 separately, and the same goes for when it's decreased. A base reduction of -1 will make her do no damage at all, but a +1 will double her output. It essentially makes damage modifiers into multipliers for her innate power, which is in keeping with Fanatic being a high-risk/high-reward character.
So if you are looking to increase the radiant damage, you're in luck. Any increase to Fanatic's damage output will increase both the melee and radiant damage.