Some languages use symbols rather than letters and scripts.
The rules don't specify a particular script or family of runes, which leaves the detail of how the secret Druidic language is presented in written form up to each DM/campaign.
The PHB text that you cited is what you have to work with in 5e, RAW. So what do you do? Tapping into previous editions is one choice.
An example of previous edition lore: AD&D 2e
Script per se may not figure into the druidic language at all. For example, in AD&D 2e, in the Complete Druid Handbook, under the heading "The Secret Language" we find:
Finally, the secret language of the druids remains a purely spoken
tongue. A few simple runes or marks (symbolizing danger, safe water,
safe trail, and so on) exist for marking paths and leaving messages,
but the language cannot communicate actual sentences and complex ideas
This points to the Druidic language as being symbol based, not script based. That has the potential of making Druid Scrolls problematic (as they were in 2e). This can be an obstacle if you want Druid spells to be available on scrolls in your 5e campaign. It might also be workable, in that the Druidic language and Druidic magic are rooted in the same basic philosophy and symbolic system. When it comes to magic, symbols often substitute for words/letters, as shown in the Glyphs of Warding in Clerical magic, for example. Druid symbology written onto scrolls can get around the obstacle and make spell scrolls viable.
The Rules Don't Try to Cover Every Granular Detail.
Where the rules are sparse, the DM and the players are expected to flesh out the detail in a 5e campaign. That's a feature, not a bug.
Recommendation: treat the Druidic secret language as symbol based, rather than script or letter based, using the guidelines in AD&D 2e as a point of departure.
Aside: historically, a symbol-based form of writing, hieroglyphics, was very effective for the Ancient Egyptians.