A garden wall carries real structural load. Footings, piers and wall ties are done properly, not just enough to pass a glance.
A typical wall job starts with protecting the area, boarding a driveway if one's in use, and putting up temporary fencing if any of the work is near a footpath, to keep residents and passers-by safe while work is underway.
The existing wall, if there is one, is taken down and loaded out, then a new footing is dug, typically around 18 inches wide and 70cm deep, filled with roughly 60cm of concrete. That gives a proper bearing either side of a 9-inch wall. From there, the wall is set out with piers built roughly every 3 metres on a straight run, tied into the wall for maximum strength, and every few courses the brickwork is tied in with wall ties for extra rigidity, more than the minimum a 9-inch wall technically needs, because it's worth doing properly.
For repointing rather than a full rebuild, the process is different: joints are ground out, then cleaned with a sodium hypochlorite wash before repointing with a 4:1 sand and cement mortar, packed in tight for strength and struck off with a pointing iron to get a clean bucket-handle finish. Steps and smaller areas are sometimes repointed without the cleaning stage, to keep them looking consistent with the surrounding brickwork.
Brick costs can vary a lot depending on the match you're after. Where a close brick match is available at a lower cost through Harry's trade contacts, he'll flag it and let you decide, rather than defaulting to the most expensive option. Materials and labour are always priced as two separate figures.