Sounds like you have a great trip planned and hope it all works out for you.
To ride a bike in the UK you need insurance, tax and MOT. For insurance try eBike Insurance - Bike and Classic Bike Insurance from eBike Insurance
, they do monthly policies which should be good for a short trip like yours, and are valid throughout the EU. I paid 30gbp per month on my XR400, i'm 25. I guess you have to phone them and check if they'd cover you with an Aussie licence. Insuring a bike in London tends to cost 2-3 times as much as an rural England, so if you have any friends out in the sticks it would be a good idea to register and insure the bike there.
The MOT is the annual roadworthiness test for the bike, buy one with at least a few months left on the certificate and you won't have to worry about it. Tax also has to be paid each year, i think it's 30gbp/year for my 400. you need to have insurance and MOT to be able to buy the tax, and if you don't have tax you get in trouble. make sense?
when you buy a bike there's no charge to put it in your name. Send off a form and you get the certificate back in 3 weeks. If you don't have 3 weeks between buying the bike and leaving the country (you need to carry the registration certificate if travelling in Europe), you can get a temporary certificate in a couple of hours by going to a DVLA office - see dvla.gov.uk
i'm not sure how much an old BMW should cost, but looking at biketrader.co.uk and ebay should give you a good idea. March is around peak buying time as people get new bikes for the summer, so prices may be a bit higher and bikes sell quickly. Also prices tend to be a little higher in London than the rest of the UK, and/or the bikes tend to be in worse cosmetic condition - it's a hard life for them commuting through london traffic everyday!