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.
The psionic power mindlink does exactly what you want: It's low level and works on any creature that has an Intelligence of 3 or higher. (As long as you spend the points to make it work on an unwilling target.)
And most importantly:
You can communicate telepathically through the bond even if you do not share a common language.
As a backup option, as long as you can touch the ice toad, you can cast tongues on it. Tongues doesn't require a willing creature, so as long as the ice toad fails its will save this should allow it to speak to and understand you.
This spell grants the creature touched the ability to speak and understand the language of any intelligent creature, whether it is a racial tongue or a regional dialect. The subject can speak only one language at a time, although it may be able to understand several languages.
This won't allow it to speak to other ice toads:
Tongues does not enable the subject to speak with creatures who don’t speak.
So if you want it to be able to speak to other ice toads, you'll have to cast the spell on them too.
There is one other fly in this ointment:
The subject can make itself understood as far as its voice carries.
This is potentially problematic, because if ice toads don't have a voice (which may or may not be the case), they will be able to understand you but not speak loud enough for you to hear. This will enable any communication method that relies on the creature understanding you, and might allow communication methods that rely on the creature having a language. (Depending on whether understanding every language counts as "having a language"; I can't find a rules definition of what that requirement actually entails.)
Alternatively, with one-way communication established, you may be able to talk it into accepting a telepathic bond or some other method that relies on having a willing target.
Script refers to the characters used. To an Orc, text written in Dwarvish would have familiar letters, but otherwise would make no sense to him.
Your comparison between English and French is spot-on: both have the same script, but being able to read English does not help you understand French. Compare this to for example English and Chinese: not only do you not understand, even the very symbols used to write the text with are alien to you.