I have milky white oil in the cooling reservoir. The vehicle in question is a BMW 318i and was bought 5 days ago.
What would be the most like cause?
Welcome to Mechanics SE! A white milky, mayonnaise like substance is indicative of oil and water mixing. This could be down to a failed cylinder head gasket or a failed oil cooler. I'm not sure if your car has an oil cooler but the first thing I'd do would be to have the engine either compression or leak down tested. Either of these tests ought to indicate the condition of your cylinder head gasket.
Textbook root cause nr. 1 would be a damaged head-gasket, other possible issues are:
Either way this is something serious, I would try to rollback the purchase (if possible) or let someone fix the issue (costly).