In slow, careful enunciation it's /'ɒbviəsli/ in General British; in colloquial speech this normally becomes [ˈɒb̪visli] or [ˈɒb̪vəsli]. In extremely casual, rapid enunciation you may hear [ˈɒvəsi] or even [ˈɒvəʃi].
None of the three pronunciation dictionaries – LPD, EPD, ODP – tells you this.
More on such chameleons in future blog entries!
Another possibility is [ˈɒvvjəsli].
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