Yes, private health insurance does cover specialist appointments. You will normally need a referral from the relevant professional, such as a GP (whether NHS or private), an osteopath, nurse practitioner, chiropractor, psychologist, or even a speech therapist. Talk to your insurer to find out which referrals they accept.
Some insurers will give you direct access to specialist appointments without a referral if your policy covers the suspected condition. For example, if you have potential cancer symptoms, mental health problems, or musculoskeletal issues, you may be able to contact your insurer direct and get a referral to a specialist.