While the NHS is funded by taxes, it is not a service whereby you can have anything you want, and never has been. The NHS does provide a range of contraceptive options, free at the point of use. If you choose to use a contraceptive method that they do not provide then I can understand why you are expected to pay for it. I would imagine that the other issues with condoms versus the pill/ coil/ etc is that who decides how many condoms are on a prescription? Do you get 3 a week? Will the people who want 7 a week complain? Do you get 7 a week, well what happens when you are on holiday and have sex more than once a day? Do you complain then? With other methods of contraception these issues do not come into the equation.
This is not restricted to condoms. When my Mum was dying, there were drugs that the NHS funded for her type of cancer. There were other drugs that they did not fund and if she chose to go down that route then she would have been expected to fund those herself. These decisions are made across the NHS at all levels and for many different conditions/ illnesses.
There is not an infinite pot of money and as people are generally unwilling to pay increased taxes, they do have to make decisions about what monies will and will not be spent on. Of course we can question why not supply condoms but for me, the NHS offer a variety of contraceptive methods to choose from and the line has to be drawn somewhere. Believe me, I do know all about how frustrating it can be when the treatment line is drawn and your life is affected by it.