It doesn't take resources from the NHS.
Consultants work their contracted hours.
What other resources are you thinking of?
Imagine you need a procedure Call; the hospital tell you that they can do 10 a week and there are 40 people ahead of you so the waiting time is 4 weeks. Or, Dr Smith at St Moneybags Infirmary down the road can do it; they have capacity for 10 a week too, but since you have to pay, the waiting list there is just 10 people so you can have your procedure in just a week! Hooray!
So clearly there is capacity to do 20 a week. But those with the means - personal or through their employer - can jump the queue. If the 20 a week capacity was available in the NHS, people in general would be seen quicker.
Now I'm sure someone will come along and tell me that's not how it works, the capacity wouldn't be there if it wasn't for private money. Well its only that way because we allow it to be, because we think that's how it should work.
I think we should pay more for the NHS too Call, whether through more tax, or a different funding structure, I don't know. But I know that being able to jump the queue and divert resources from people who are not as well off as you is no way to run a Health Service.