Cool maps showing an unsurprising result
mySociety.org have some cool maps showing travel times for central London. All maps show travel times to the the Department for Transport (DfT) office in central London.
This one shows areas where going by bike beats public transport if you need to get to the DfT office (note that all the areas that are not in grey are the ones where the bicycle is faster):

Some rights reserved (based on OpenStreetMap)
This is not surprising and matches my experience. It is still impressive, especially to see how far the area extends. I’d add two caveats, though.
Don’t misread the map as presumably saying that all the trips within the shown area are faster by bike. Remember that we are talking about trips to the DfT office, shown in the centre. Additionally, even if you take other trips in central London of about the same distance, if your trip just happens to conveniently start and end at a directly connected tube station, the tube would win hands down.
I’d also like to know the cyclist’s speed that was the basis for the calculation. I’m able to go pretty fast (as far as traffic allows it), but I’ve found the times given by the DfT’s own Journey Planner when selecting the “cycle only” option to be wildly optimistic.
Still, I can confirm that from Hackney (where I live) to basically any location in central London I’m on average faster by bike. I’ve actually done quite a few involuntary experiments when I was with someone who didn’t have a bike and we needed to do the same trip. In the best case, when the bikeless person got lucky and caught all buses without a wait, we arrived at about the same time. In most cases, I was between ten and twenty minutes faster. In one case, I was an hour ahead (I don’t remember exactly what that was — some extreme situation like a bus line not running at all, I guess. In all fairness, the equivalent scenario for this would be having a puncture on a bike. I often don’t carry a repair kit.)