For instance, if a mobile phone provider received a court order/request for subscriber details because of ‘illegal’ activity, would they cut that subscribers services because of it or even make their connection slower?
Whether or not a provider will cut off or restrict service depends on (a) what the contract says and (b) whether the company is willing to ignore the contract. If they ignore the contract, and the contract doesn't allow them to cut off service in that circumstance, they may get sued, but that depends on the customer. No principle of law say that if a company receives a court order to produce data pertaining to a customer, the provider is compelled to terminate service (a court order could conceivably itself mandate terminating service). Without specifying a jurisdiction it is hard to say whether it is true in all countries, but in no jurisdiction that I know of is a provider entitled to breach a contract because the court has issued an order to provide information about a customer. In general, if you want to know what a phone company can do, read the contract.
It is routine for contracts with service providers to contain provisions where the customer agrees not to use the service for illegal activity, and it is routine to include provisions allowing the provider to take certain actions as a result.
I don't know anything about your mobile contract, obviously, but I can tell you that AT&T, for instance, will slow down service for people who are repeatedly the subject of DCMA infringment notices, and that they at least threaten to cut off service, as well. Don't ask how I know that.