Plot: Learn it from someone who knows it
One would presume that should a druid decide to teach you Druidic, then you could spend a skill rank or two on Speak Languages to learn the language. That would be a DM’s call and probably quite rare, but it could happen. A blighter (Complete Divine) certainly wouldn’t care about teaching the language, for example.
Mind reading, memory access, or simple eavesdropping on a druid teaching a new druid may also suffice to be allowed to use Speak Language to learn Druidic. Note that the druid may lose his class features even if you force or trick him into teaching you though.
And of course, if one can find a nondruid who knows the language, that person would have no reason to refuse to teach you.
Magic: Speak all the languages
There are magical means to understand languages; comprehend languages and tongues would give one the ability to temporarily understand and speak Druidic.
A crystal mask of languages would also be capable of doing the trick. Oddly enough, while the crystal mask of languages requires that the crafter speak five languages, and then grants knowledge of five languages, nothing says they have to be the same five. Thus a psion who knows several languages could craft one that grants knowledge of Druidic.
Note that none of these actually teach you the language; they all last only as long as the spell does, or as long as you wear the item.
Epic: Polyglot feat
The [Epic] feat Polyglot says you know all languages, so that includes Druidic. Language barriers are probably not your greatest concern at level 21, though. I suppose some Dragon abuse could get you the feat earlier, which in this case wouldn’t even be game-breaking.
Prestige: Loremaster or Race: Tibbit
This is the most useful one: the loremaster prestige class gives bonus languages at 4th and 8th level. This class feature simply states that the loremaster can choose “any new language”—it doesn’t exclude Druidic like racial bonus languages do, nor does it invoke the Speak Languages skill.
Likewise, the tibbit race from Dragon Compendium includes “Bonus Languages: Any. Tibbits travel far and wide and their curiosity pushes them to learn a number of languages,” in contrast to the human entry of “Bonus Languages: Any (other than secret languages, such as Druidic). See the Speak Language skill.” Again, no prohibition on secret languages, no reference to the Speak Language skill.
Either of these could be oversights and errors, but neither has been corrected (and both books have seen errata), so I for one am willing to take them at their word.
And while Druidic itself does say that it is “a secret language known only to druids,” 3.5 has a very important rule known as “specific-trumps-general.” In this case, I would argue, the general rule is that only druids known Druidic. After all, they get it free and everyone else has to work quite hard to get it. But clearly it is possible to learn the language in general (see previous points), so the statement that only druids know it is not absolute. Polyglot specifically supersedes it. So, I would argue, does the loremaster’s or tibbit’s bonus languages. However, this is only a case I am making: you could argue the reverse, that the Druidic rules are the more specific. You will have to discuss it with your DM.
Thieves' Cant isn't a written language, thus there would be nothing to understand via a spell.
Nowhere in the quote you've pulled (or the PHB) is thieves' cant ever described as a written language. This is because thieves' cant is both verbal and physical communication. Some word substitution (1 to 1) is used, but it is largely based on metaphor and contextual meaning and a big part of this is the hand symbols used when speaking. D&D's basis for thieves' cant is both historical and a trope.
The symbols mentioned are more like pictographic signs than words.
As such they are not translated, but identified, similar to how we use symbols such as the biohazard sign and nuclear sign to signify specific danger or how the symbols on a crosswalk signify when to wait and when to go. The closest living example of this I can highlight would be Hobo symbols that survive and are still in use today in the US. Different symbols would mean different things to different groups and insider knowledge for understanding thieves' cant symbols would be a must.
Best Answer
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:
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.