I think in order to answer this question we need to figure out what exactly it means to “read minds.”
Arguably, we can already do that with fMRI and EEG. (An fMRI estimates what parts of your brain are active based on changes in blood flow. An EEG measures electrical activity in the brain.)
I could use an fMRI to tell you which image out of a set of stimuli you’re looking at with pretty good accuracy. (This study found that you could do this with about 90% accuracy using a set of 120 images, and 80% accuracy using a set of 1,000 images.)
So that kind of mind-reading would (and does) transcend language. But it sounds to me like what you’re really interested in is reading something like the engram, which is a hypothetical memory unit in the brain. We don’t really understand what these might look like, or if they even exist. Most neuroscientists believe that a memory is an emergent property of the activation of neural networks. It seems to me that some of these engrams or neural networks use language, and some don’t.
I would hypothesize that most engrams, to the extent they exist, are probably the summation of activating connected, recorded perceptions. Most of our perceptions are independent of language but could be recorded as language if we chose to articulate them. (This is my speculation, not real science.)
I don’t think anyone could argue that our brains require language to record memories—otherwise most animals would be out of luck, and I’m pretty sure I can give you evidence that every organism with a nervous system has at least some kind of memory.
I don’t know if this really answers your question, but I think it may be unanswerable by science (for now).