Disadvantage of oversampling
First thing: the Nyquist rate is not sufficient to obtain a correct sampling of a signal, it's just the theoretical minimum. Reasonable sampling rates go from twice the Nyquist rate (four-five times the maximum frequency) up.

Several ADC architectures use oversampling with averaging to obtain higher precision than the converter itself achieves. The extreme case is the one of sigma-delta converters, where a 1-bit ADC (just a comparator) is run at very high speed (\$2^N\$ samples/value, where N is the resolution in bits) to achieve the highest linearity, because the 1-bit conversion is linear by definition.

The drawback of oversampling is of course higher speed required for the ADC and the processing unit (higher complexity and cost), but there may be also other issues. You can see also that, at a given ADC speed, oversampling will require more time so an overall slower speed.

Another possible drawback is that it may result in additional noise if, for instance, the lower sampling speed allows you to integrate on a longer time.