Depends what is wrong with you.
Mostly there are "rapid access" schemes for chest pain. When I started with angina I had blood tests and ecg ordered by gp straight away. When she got the results a week later and they showed problems I was passed onto the local hospital. Within 2 days I was called in for an exercise ecg. That's where the "rapid" stopped for me.
Exercise ecg was +ve and I was given an appintment with a cardiologist but that was
7 weeks away. After that appointment I was told I needed an angiogram. I waited
5 1/2 months for the angiogram which showed that I had a 95% blockage and a 90% blockage in 2 main arteries

. They got me in just over 3 weeks later and I had angioplasty with 3 stents. I don't like to think too much about the fact that I was walking around with those blockages for all that time. However I was on medicine i.e. aspirin, beta-blocker and nitrate spray so that kept me going.
In some ways I wish I had speeded up some of these stages, especially initially, but of course these tests are expensive. I think, looking back now, it would have been worth me paying to see the consultant and also maybe paying for an earlier angiogram. In most specialties one can see a consultant privately within 2 weeks. Things are better now with the NHS than they were years ago I believe. Don't get me wrong - I am so impressed with the care I have had overall and with what has been done for me on the NHS, and of course any waiting time seems like ages when one is the patient.