There is a group of people who are effectively invisible, functioning adults with no language. They are there but we just not not met them. They are born completely deaf and are not taught sign language or lip reading and, in fact, miss out not just on language but on knowledge that language exists. Now that they are known to exist, the question arises, how? Those non-linguistic adults live amongst us without being noticed (unbelievable - wild animals live amongst us in our cities and most of us do not see them). It must be much harder to survive without language than with it. So we must accept that these people are very good at understanding and using their environments. They must be continually solving problems -successfully. No sheltered workplaces, social workers, welfare payments, special education or any aspect of the net that is meant to catch the handicapped is available to them. No help is available from all the written and verbal signposts that litter our streets and airways. They cannot talk with others to ask or tell anything. They survive presumably because they are very intelligent, continuously observe the world and use their cognitive abilities to their up most. A description from neuroanthropology is (Life without language) and I urge you is read it.
So can people have thought without words? Well, the evidence-based answer would seem to be, yes, but its not the same sort of thought. Some things appear to be easier to get without language (such as imitation of action), other things appear to be a kind of all-at-once intuition (such as suddenly realizing all things have names), and other ideas are difficult without language being deeply enmeshed with cognitive development over long periods of time (like an English-based understanding of time as quantitative and spatialized). In other words, language is not simply an either/or proposition, but part of a cognitive developmental niche that shapes both our abilities and (unperceived) disabilities relative to the fully cognitively matured language-less individual.
Here is my take on the difference between cognition with and without language absolutely speculative exercise in guesswork take it with a grain of salt.
Problems that involve only sensory precepts or motor actions can be solved with or without consciousness either way language is not needed. So our language-less man would be aware of his surroundings and his intent/action arcs like an ordinary person and would have memory of that awareness. To this extent his consciousness and his cognition would be like ours. He would even be able to manipulate some concepts or symbols although it is questionable how abstract these non-linguistic concepts can become. We can assume what language is not required for much of the simple communication between people. If other primates can live their lives without language why should a human not be able to do it.
But there is two sorts of thinking that I cannot imagine a language-less person engaging in. This is the kind that uses the cycle of: taking to yourself, being conscious of the inner voice, holding it in working memory, using access to that memory to retrieve the idea in the inner speech. This cycle would allow two parts of the brain that are not well connected in the manner needed, to exchange information through the global access available in consciousness and working memory.
The other type of thought that might be difficult to the person without language is the elaboration of abstract concepts. I believe this depends on nested series of metaphors/analogies. As the child metaphors become more distant from their concrete original parents, they become, in effect, a set of symbols related by a set of relationships. The connection to the senses and actions are lost. Without a ‘language system’ it becomes more and more difficult to handle more and more abstract symbols and relationships. I assume it would only be possible at an elementary level.
We know that handicapped people find ways around their handicaps and so I would expect the language-less to be very resourceful in developing ways to think that bypass the need for language and this might actually make them better at some specific cognitive tasks. But there is a limit.