Language has been called a cognitive or cultural tool, a description that succinctly summarises the research on the effects of language on cognition and the mind.
This research also answers your question with a definitive yes: a broader vocabulary can allow you to think faster. Assigning a label, which is basically what a word is, to a concept, allows it to be used more easily in your brain.
Consider the evidence: one study looked at differences in counting ability and counting memory between speakers of English and speakers of Piraha [an Amazonian tribe]. The Piraha language is notable for many things, one of which is its lack of any words for numbers or counting. What they found is that the Piraha can still recognise quantities – not surprising. However, their memory for quantities was worse than English speakers'.
Not only is the existence of a word important, but so, too, is how easily you can access it.
In another study, based on Daniel Oppenheimer's work on processing fluency, the researchers looked at short-term stock-price fluctuations and its relationship to the stock ticker symbol (eg, MSFT for Microsoft, FB for Facebook). What they found is that stocks with symbols that were easier to say did better immediately after IPO [Initial Public Offering].
And riffing off that general idea, I worked with some colleagues looking at the difference in hard-to-pronounce and easy-to-pronounce plural words (eg, 'keys' as easy, versus 'busses' as hard). What we found is that the plurality of objects for easy-to-pronounce plurals was remembered better.
One thing to note: most of what I'm talking about here is accuracy and you're talking about speed. In cog sci, the two are often interchangable because of a well-known speed-accuracy trade-off.
What may be surprising is how quickly these effects begin to happen. Recent research has shown that when learning a new word (in your native language), the new word begins to affect cognitive processing as quickly as a day later. This is particularly true if you've had a chance to sleep on it.
Ultimately, your question serves as a useful companion to another Quora question: does an increased vocabulary change the way you think?
In answering, I suggested that there is some effect of an increased vocabulary, but 'change the way you think' is a bit of an extreme way to put it; vocabulary doesn't serve as a straitjacket for what you can and cannot conceive of. Almost every invention was an idea before it was a word, for example. But there is definitely an effect, as the answer to your question shows.
Marc Ettlinger, PhD Linguistics, UC Berkeley