Unfortunately, at the assessment centre you simply won't be afforded the sort of arrangement you have at your GP's surgery. For the ESA assessments, you pretty much have to sit there and just wait for as long as it takes to be seen regardless of how much pain that causes you. Of course, you should tell the assessor when you eventually go in what the effect of having to sit for the duration of the delay has been on you as that is relevant to your claim.
As Monic says, its only if/when they turn round and tell you that they will not be able to see you that you can leave without the risk of being deemed not to have attended the appointment. If you try to tell them that you need to leave because you are unable to wait for the duration of any delay, you are likely to end up in real difficulty with continuing your claim. That will especially be the case in the event of a very minor delay like 10 or 20 minutes. I know that is a long time to you but in the world of these appointments, its nothing at all so it wouldn't be considered reasonable for you to ask to leave after such a short delay.