I'm assuming you're covered for NHS care with your EEA residence card?
First thing you need to do is register with a GP (if you haven't already), then make a free NHS appointment with the doctor. When you go to the appointment, tell the doctor what is bothering you and ask if he can do some tests. If the tests are covered on the NHS, they will be done for free and you won't need to worry about private insurance or paying for anything.
If it turns out that they aren't covered by the NHS, then you can ask for a referral go to private... and I think it's only when you have a referral to a private doctor or consultant that you need to bring up any private cover you have.
A few years ago I was feeling really run down and tired and my mum thought it might be anaemia. So I made a doctor's appointment, told the doctor my concerns and he ordered a blood test to test for various things...I made an appointment with the nurse for a blood test, had the test, got the results (turned out nothing was actually wrong) and went home... I didn't pay a penny as it was all on the NHS.
Private insurance generally isn't required for most things in the UK as a lot of treatment and tests are covered by the NHS...usually people only use their private cover for extra things that either aren't covered on the NHS or to skip waiting lines for treatment... for example, my mum has used her company health cover to pay for some physiotherapy session for a bad knee, and her friend paid out of pocket to get a hip replacement done much quicker than it would have been on the NHS (i.e. only a 3-week wait for the operation vs. 3 months).