As you have already hinted at, the issue is controversial. I could leave it at that and say "no, there is no consensus", and it would be a true answer, but it wouldn't be satisfying, wouldn't it? Instead, I'll briefly define the topic, give a few examples, and then a few recent criticisms. My answer will be weighted somewhat towards "cognition" instead of "computation", however different you consider them. It's more of a safari than a good answer I fear, I hope you're getting something out of this.
Neural synchrony is usually used to refer to oscillatory alignment; that means, some sort of oscillation occurs in a population of neurons, for example, an alpha rhythm as a ~10 Hz pulse; and neuronal synchrony is both, on the individual level, the degree to which individual neurons either take part in this oscillations, and/or align their spiking behaviour to certain parts of this oscillation, and on the population level, how other neuronal populations are aligned with this oscillation.
What is doubtlessly true is that the brain is full of oscillations. We know that at the very least since Hans Berger discovered the alpha wave back in the 1920s. When someone closes their eyes, we quickly see the 10 Hz rhythm. But once they open the eyes, or are disturbed in any way, it quickly disappears. Are these oscillations important per se, or just an epiphenomenon? Berger himself assume them to be the direct manifestation of unified brain state. But Berger was a bit of a crank. The well-respected British researcher and Nobel price laureate Lord Adrian who first replicated this finding assumed the 10 Hz rhythm was simply the "idling of the brain", the very absence of computation.
One contemporary researcher who assigns a functional and active role to the alpha rhythm is VanRullen, and I think much of his work is very interesting.
Let's look at some of the rather basic ways how oscillations and timing are relevant in the brain. First, timing in general. Our original understanding of neural communication (in part unveiled by aforementioned Lord Adrian!) focuses on the concept of rate coding and population coding; the net amount of spiking activity in a neuron, or in a population, is what matters. This is for example what happens in the peripheral nerve cells; when heat or pressure or pain are applied to the skin, the responding neurons directly communicate the degree of heat or pressure or pain by how much they ramp up their firing (at least at first, before Spike Frequency Adaption sets in).
Initially, this was seen as fully consistent with Hebbian learning - the more one neuron fires, the stronger the synapses with its targets become. But then Bi & Poo discovered Spike-Time Dependent Plasticity (STDP), and now everything is a bit more complicated. Remember Hebb's Law says
When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A's efficiency, as one of the cells firing B, is increased.
Hebb's Law goes beyond saying A has to fire lots. It says, A's firing has to cause B's firing. And one component of causation is temporal precedence; so A's firing has to occur before B's firing for increasing synaptic strength. And this is just what Bi & Poo observed; only when A's spikes arrive in a short temporal window before B's own spike does Long-Term Potentation occur. If however B's spikes happen a bit before A's spikes, Long-Term Depression occurs.
So the temporal order of spikes matters. Neurons causing others to spike gain more influence over them, and neurons who're followers, not leaders are demoted.
Another bound for plasticity is of a neurochemical nature. Levels of catecholamines (neuromodulators) regulate if the hippocampus will encode an incoming stimulus, depending, for example, on its novelty. So only if neuromodulator levels are up does incoming information lead to lasting adaptations. During sleep, were we assume reconsolidating to happen, we observe that the brain stem sources of catecholamines (like the neurotransmitter norepinephrin and dopamine) are temporally correlated with the slow wave occurring during sleep. Specifically, Locus Coeruleus neurons spike right at the onset of the transition to the Up state. It seems the cortical oscillation is timed so that active states co-occur with high catecholamine levels.
This is a form of temporal coding called phase coding. The phase of the ongoing oscillation matters with respect to the incoming spike. It is also observed, for example, in the theta oscillation in the hippocampus. A range of theories exist for a possible function of theta phase locking; examples are that certain theta phases again indicate more receptive states (so that incoming information has a higher chance of being encoded during certain parts of the theta oscillations), or that the part of a theta phase a neuron spikes at indicates how important the information this neuron is encoding is.
Beyond phase coding, we find Singer's very popular binding by synchrony. How do to parts of the brain communicate to each other that they are "talking" about the same thing? Well, by alignment in the gamma band, that is, by phase locking of the two populations to the same ~40 Hz rhythm.
But how believable is all this?
There are many theoretical objections to specifically high-frequency binding by synchrony. For example, the power of oscillations fall of over distance as a function of its frequency; that means, slow oscillations travel far, but synchrony in fast oscillations cannot be sustained over long distances. (Slow oscillations are simply too slow for binding by synchrony.)
Another question is if there are even any gamma oscillations at all, if they are truly clock-like wave patterns or if they are simply a methodological artefact - traces of the sensitivity of the Fourier transformation to asynchronous activity, or filtered noise.
At least for humans, gamma band synchrony findings have become strongly suspect as it was shown that these patterns are better explained as the EEG traces of small eye movements or micro-saccades.
But this must not spell the death of all theories for a functional role of timing and oscillations, of all temporal codes and phase codes. In STDP, we have clear, anatomically sound evidence for the importance of temporal order in spiking, and simple evolutionary principles suggest neuronal populations will exploit such phenomena. However, the field is diverse and controversial. There is no universally accepted phenomenology, methodology, or theory. We have tidbits and fascinating findings, but nothing coherent. Cortical oscillations are sill mostly a riddle.
I think most recent contribution is the discovery of Cross-Frequency Phase Coupling by Canolty et al. They observe that the amplitude of the fast (gamma) oscillations depends on the phase of the slower theta or delta rhythms. The basic finding is quite similar to the observation of LC spike/cortical state phase coupling noted earlier, but now we observe this in the awake animal. And a clear proposal for a functional role of this mechanism has been made, too; I can simply quote the abstract of Lakatos et al.:
Whereas gamma-band neuronal oscillations clearly appear integral to visual attention, the role of lower-frequency oscillations is still being debated. Mounting evidence indicates that a key functional property of these oscillations is the rhythmic shifting of excitability in local neuronal ensembles. Here, we show that when attended stimuli are in a rhythmic stream, delta-band oscillations in the primary visual cortex entrain to the rhythm of the stream, resulting in increased response gain for task-relevant events and decreased reaction times. Because of hierarchical cross-frequency coupling, delta phase also determines momentary power in higher-frequency activity. These instrumental functions of low-frequency oscillations support a conceptual framework that integrates numerous earlier findings.
Here is a video presentation on the topic.