In the UK, it is possible to have IVF treatment on the National Health Service, in accordance with guidelines from NICE, the National Institute for Health and Clinical Excellence. The waiting period may vary, and up to three cycles should be available on the NHS if you meet the requirements that include being of a certain age and having an infertility diagnosis.
There may be additional requirements like stopping smoking or having no kids yet. Private fertility clinics should be licenses by the UK fertility watchdog HFEA, the Human Fertilisation and Embryology Authority. The success rates of the clinic you are looking into should be available from them, as well as the HFEA. Figures from the same organization have shown that London fertility doctors have better success rates for IVF than doctors in any other part of the United Kindgom.
Interestingly, the Assisted Reproduction and Gynaecology Centre in Upper Wimpole Street was shown to have the best success rate; more than double the national average for live births following fertility treatment. Their Doctor Mohamed Taranissi had the most babies born each year. That's something you may want to look into if you are in the market for private fertility treatment, but no one doctor or clinic is right for everyone who needs fertility treatment. If you'd like some more general ideas on finding the best clinic for you, read How to choose a fertility clinic.