You are able to change your health insurance provider however there are a number of points you should consider before doing so.
Many insurers will let you switch to their products on a continued personal medical exclusion basis so you can continue your cover on the same terms as your existing cover. It is important to ensure the provider you move to will continue cover on the same basis otherwise medical conditions which may be covered under your existing plan may be excluded moving forward.
Naturally there are a vast range of health insurance products on the market and no two policies are the same. To ensure you are comparing apples with apples it is important to compare the benefits of each insurer, compare any cover limits or monetary caps on treatment and get a good understanding of the benefits and limitations of the alternative options.
Private medical insurance is an annual contract and thus if you are going to switch insurer it makes sense to do so at the policy renewal to ensure no payments would be lost.
As with your initial application for a health insurance policy there will be paperwork to complete and it varies from insurer to insurer. When switching health insurance provider it is likely the application will be shorter and they may request a copy of your existing policy certificate as proof of your existing cover.