As the meter works on other ranges, I suspect the rotary switch. When the meter is switched to the 5A range it connects the movement and a shunt resistor. If the contacts are dirty, worn, or bent the selected range won't work.
This circuit is typical, note the rotary switch uses two poles for current ranges.

Not familiar with the GPO meter. Some meters I've been inside are easy to trace and test, others hide stuff behind layers of other stuff. Rotary switches are often built-in, part of the PCB, and not replaceable. However, my experience is mostly with inexpensive meters, designed on the assumption they won't be maintained: glue, rivets, plastic clips, non-standard parts, no circuit or service manual etc, etc. A GPO meter built for solid reliability is more likely to be fixable than a cheapo meter, but much depends on the way it's built.
Can you post a photo or two of the innards?
Dave