Thats what I have done / am doing
I have to say... changing to a scooter in London is the best thing I have ever done for personal transport. It literally costs pennies to run, and is just as quick, if not quicker!
Its more civilized, more dry, more comfortable, less stressful (I can focus on the road rather than fidgeting about with gears in heavy traffic) and I can actually carry my kit and shopping inside the luggage space on the bike, and I do not have to dress up like a power ranger either
For you, I recommend something with a bit of grunt though - and not the smaller European scooters (Vespa, even the super, would be a bit too small I think, even though its nearly as nippy as my scooter).
I recommend slightly larger wheel sizes because London roads are appalling, and larger wheels handle potholes better.
Apart from the fact you get 3 to 4 times better mpg than a Rocket 3 on a scooter, that is not the only reason why not to commute on it. Its like using a nuke to kill a fly, using a rocket 3 for commuting! How soul destroying would that be - looks good, but just not practical like a scooter.
Do both - Autotrader will see you with a yamaha majesty 400 or a used T max for about 2-3000k - or an older silverwing (650 cc) if you fancy a lardy arsed slow fat huge thing. The 400cc Suzuki burgervans are ok, but the bigger engined ones are pointless and heavy for traffic work... you want something mid sized, to cut through traffic easily. And a minimum of a 250 - 300CC engine - preferably 400cc is the sweet spot. The TMAX yamaha handles close to a sports bike, and is quick and a joy to ride, but the mpg is not overly good - 45-50mpg only. Step a little bit down, and 400CCs will see you up at the 60-80mpg range - and its very nearly just as quick. Commuting is all about comfort, weather protection, ease of use, dependability, and ability to cut through traffic - none of which remotely apply to a Rocket 3 (relatively!!!!!!).. which is just overkill.
Brands I recommend - Japanese, like the majesty 400- some piaggio group ones are a bit less well put together. Alternatively, you could always go the small-mid sized thing like me, but I would be less comfortable on motorways due to the presence of it being less than the maxi size scooters.
It also means less mileage and wear and tear on the Rocket 3!