Personally I think the police knew the phone pinged Milton at the same time they found out it pinged BM. The data would all be together from the network provider.

The big question now is as you say, what time did it move from BM to Milton? But the answer might not matter as:

I guess the police still think it was unlikely that C's body could get through the processing plant and onward landfill so pursued other avenues for his body regardless of the fact the phone could have ended up in landfill via HS bin lorry.

So they conclude that C and the phone were separate from each other and therefore a third party involved. Faced with the choice, the police spend the limited time and money on gathering information about a criminal possibility rather than waste many weeks searching landfill and 'freeze' the evidence in that landfill for a later date when all other enquiries are exhausted or fruitless.

Does make sense now and does confirm either they have run out of worthwhile possibilities to spend the budget on or now think the search will provide enough evidence to turn it into a murder enquiry and gain further funding or close the case.

The state of decomposition by this stage might render cause of death as inconclusive but finding the phone with him may help give further information.