Well the question is why would the British want them? After 1867 they are a money sink, hence the Danish were willing to sell as early as 1867 but didn't pan out due to US politics, again it was tried in 1902 but Danish nationalism stopped it. The US was interested because they wanted a base in the area, whereas the British already had that, so why do the British want a money sink?
The US finally bought them to keep them out of German hands as a submarine base, this was okay because both nations were neutral at the time, whereas Britain was not, the purchase had to be rushed so it would occur before the US entered WWI
So you need the British to really not want someone else to get them, be willing to outbid the US and not trigger Danish nationalism, and do it before WWI or equivalent breaks out