There are different levels of training and authorisation as mentioned regarding the use of blue lights. It depends on the force and their policies. Some may be allowed to use the blue lights to ask a car to pull over compliantly, but not allowed to use the blue lights in any high speed driving or pursuit. ( perhaps the case exampled by the OP ).
Some will be allowed to blue light drive at higher speed to emergency calls as required, to pull over cars and perhaps to initiate a pursuit but not continue being part of it once fully trained / authorised pursuit drivers are present.
Some will be fully pursuit trained … like you see on interceptors or whatever.
As mentioned above, in addition to that, pursuits generally need authorisation/permission to continue from supervisor control … a decision which is constantly reviewed based on the conditions, risks to others etc etc.