Even if this data (date of birth, postcode, etc) would be removed, it would still be rather trivial for e.g. an insurance company to match these records to their customers. Given one or two insurance claims for doctor's appointments, matching the medical records for identical dates is enough to pinpoint a person with adequate certainty.

Then the insurance company can grep the records for mentions of smoking, drinking, drug use or injuries related to dangerous activities like riding horses, motor racing or skiing. Now that the insurance companies can identify patients in these risk groups, they can proceed to doubling the insurance fees of these people.

I think that the insurance companies should not be able to access medical records like this.

edit: please excuse my ignorance on the UK health care system. In my country, you pay a fixed fee when using public health care and you can claim insurance on that.