Slightly confused information on Timatic, but since a GB passport does not require a visa to enter Qatar for a short stay, I see no problem.
TWOV (TRANSIT WITHOUT VISA):
VISA REQUIRED, EXCEPT FOR HOLDERS OF ONWARD TICKETS FOR A MAX.
TIME OF 24 HOURS.
- HOLDERS OF A PREARRANGED STOPOVER PAID BY CARRIER (STPC)
ISSUED TO THOSE IN TRANSIT FOR MIN. 8 HOURS AND MAX. 24
HOURS ARE PERMITTED TO LEAVE THE TRANSIT AREA, IF HOLDING
PASSPORTS VALID FOR AT LEAST 3 MONTHS AND SUBJECT TO
APPROVAL FROM IMMIGRATION.