How much would the insurance be for a motorcycle or scooter worth less than 4,000GBP in London?

I'm hoping that it would be a lot less than a 50cc scooter in Dublin worth about 400GBP costing 200GBP for insurance per year.

No one can tell you this on this site, as there is not enough information. Insurance quotes take into account information such as, size and cost of the scooter, the age of the driver, where the driver lives, where the scooter will be parked overnight, if it has any security systems, if there are any points or penalties on the driver's licence, if the driver has any no claims bonus from another insurer, if the insurance is to be third party or fully comprehensive, if there is to be an excess and if there have been any motoring convictions or accidents within a certain period of time. Only when a broker has all this information, can you get a decent quote. You could try the internet and search for One Quote Direct, they are a broker and I got really good fully comp car insurance through them with a really good insurer. Other places to try are the AA, Direct Line, or something like Moneymarket.com.