An angle grinder with a carborundum disc, they make them in different thicknesses, get a thick one, around 6mm, you'll also need a pick similar to this:
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any stubborn parts, and for parts inaccessible by the grinder...the picks aren't exactly like this, they're usually double pointed, but this was the nearest pic I could find.
You'll need goggles, a dustmask and protective clothing, start at the top of the wall and work downwards, do all the vertical joints you can reach without moving, then the horizontal, working methodically like this is the easiest and quickest way with pointing/raking out...don't start repointing until it's all raked out, and more importantly, wash each wall down with a hosepipe before starting the pointing, this will remove any dust/debris in each joint and make pointing quicker and easier, in summer you can do this immediately prior to pointing to kill the suction and improve adhesion, in winter it should stay damp for a week or more anyway....you're aiming for between 12-20mm depth on a crumbly existing mortar, and 5-10mm on a hard mortar, the latter is more for decoration than anything else.
If you've never done any pointing before, when you start, you'll soon realie that pointing trowels are manufactured incorrectly, inthat they are pointed, you'll need to grind the point off so that the end of it is rounded, about the contour of a 10p piece....if you are making your own hawk (or handboard), use a thin piece of timber for the handle because a wider one will give you severe cramp, a piece of brush steel is perfect, also a small section of foam rubber or flat foam cushioning where the handle meets the board will make things more comfortable during long boring hours pointing, as will a radio.
I've used the one you quote from Axminster (though I got it from Screwfix !) when I had to 'let in' 25 foot of lead flashing when I built an attached garage. Absolutely brillient tool on my 1905 lime mortared but cement pointed flank wall - took out a suitable depth without damaging the bricks as I had feared it might. As I have a valley against the house access was very limited hence not fancying using a lump hammer and bolster. On an SDS drill on hammer only it was a piece of cake.
In the 80s we (Spouse, me and two teenage sons) raked out and re-mortared a large semi in Bournemouth using only hand tools. It was very satisfying and I wouldn't like to use a power tool for the job in the future.
Bricks do decay, mortar crumbles at different depths, I believe that using 'feel' is gentler and kinder to the structure than going for speed.
I would say that generally if it needs a power tool, it doesn't need doing. There are exceptions (like where some clueless person has pointed with pure cement, or a very strong mortar which has shrunk and cracked).
I use a tool which looks like the front wheels from an old roller- skate, with a handle and a replaceable nail which you set the depth you want to rake out to.
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