Hospital choice for referral

[Pery]

My GP has referred me to the local hospital A to see a gastro specialist. According to the MyPlannedCare website the average wait is 36 weeks and some wait 55 weeks.
Another hospital B not far away has much shorter waiting times. Am I allowed to ask for the referral to be sent to hospital B? Woukld there be any pitfalls?
It's already gone to the first hospital so I don't know if it can be changed, no-one asked me my preference.

A bit late now for you, but we can usually scroll choices and tick the boxes for faster times or nearer locations.

@Penkie the difference is 3 months so still worth changing if it's possible?

Update in case anyone else has this issue.
The referral goes to triage before you are added to the waiting list for an appointment. A way of massaging statistics no doubt as the clock doesn't start ticking until it's been triaged. My GPs secretary was able to see that no-one at hospital A has looked at the referral yet and therefore she could switch it to a hospital B.
The unintended consequences of targets.

Thanks for the update. It is very useful