The question you ask is the topic of a very active line of research.
First you are right that light (~300.10^6m.s-1) and sounds (~350m.s-1) reach an observer at different times, as a function of distance. You are also right that visual and auditory inputs are processed at different speeds by the brain, with auditory signals being faster. The auditory system is much better, in general, at processing things quickly. Take for example the fact that the auditory system processes sounds between 20 and 20,000Hz (the auditory bandwidth), while the visual system saturates at ~30Hz (which is why movies are at 60Hz). It doesn't necessarily means you get "aware" of sounds faster than vision but this is true as well. You can react to sounds in 50-150ms, and visual stimuli in 150-250ms.
So how does your brain fuses sounds and vision, and do you get "aware" of this percept relative to "real" time? Your brain needs to make an inference, that is to guess whether a sound and a visual input are generated by the same source. Your brain uses the probabilities of sounds and visual inputs to co-occur to guess whether they should be fused or not. This is called causal inference (scientific review paper below). If a visual and auditory inputs are too discrepant in space (where they come from) or time, your brain will assume they have different sources. If they more or less match, your brain will assume they come from the same source and combine them, which provides some advantages to locate that source, for example, as you now have 2 pieces of information. When you become "aware" of the sound/visual input/combined percept, relative to the "real" world is as far as I know unknown and difficultly testable (how would you estimate when some one becomes aware of an external stimulus when all you have are external stimuli to compare to). This can probably be addressed but as I said it is an ongoing field of research (below a review on the topic).
There is a phenomenon you will probably find interesting (original paper below). Because the brain uses the general statistics of the world to decide whether to fuse or split sound and vision, it is possible to change that. Some scientists have adapted observers to a specific mismatch between sound and vision. After some time of observing a consistent delay, let's say a beep happening consistently 100ms after a flash, the observers were asked to tell whether a beep and a flash happened simultaneously. What scientists have found consistently (there are many studies looking at this phenomenon) is that before adaptation people are mostly accurate. But after adaptation the observers report (in that example) the flash as happening roughly 100ms before the beep (as they were adapted). There are some limits to that, people can't adapt to too big a delay etc. But it is a robust phenomenon. If you are old enough to have experienced bad streaming services, where sound was often mismatched to images, you will remember thatafter some time you got not so annoyed with it. This is the reason, you just adapted to the delay. Scientists have also tried to see if this is distance-dependent but results are still inconclusive (different studies found contradictory results so far).
Shams, L., & Beierholm, U. R. (2010). Causal inference in perception. Trends in cognitive sciences, 14(9), 425-432.
Hanson, J. V., Heron, J., & Whitaker, D. (2008). Recalibration of perceived time across sensory modalities. Experimental Brain Research, 185(2), 347-352.
Sugita, Y., & Suzuki, Y. (2003). Implicit estimation of sound-arrival time. Nature, 421(6926), 911-911.