Re-pointing house walls

Over the years I have gradually noticed how bad the brickwork and pointing is on my house. Some bricks are very badly spalled or damaged, and I took matters into my own hands and re-did some localised repointing when rain repeatedly came in through the top of one of my windows during those vicious horizontal rain showers.
The house is a 1920s brick end terrace, it seems to have smarter bricks on the frontage, and crappier bricks on the end and back. A neighbour was having his repointed and I was about to get the builder to do mine as well, but I did more research and got the impression that lime mortar would be more appropriate. I've cancelled and got a quote from a lime mortar pointing specialist, but it will have to wait until next summer. (Not confident next door's builder knows what he's doing with lime mortar).
Am I being ridiculous? A lot of houses around here seem to have been repointed with standard cement mix. Not all of them have spalling bricks. My joints seem to be lime mortar (soft crumbly yellow material) with a smear of brittle grey cement mortar over the top.

The house is a 1920s brick end terrace, it seems to have smarter bricks on the frontage, and crappier bricks on the end and back. A neighbour was having his repointed and I was about to get the builder to do mine as well, but I did more research and got the impression that lime mortar would be more appropriate. I've cancelled and got a quote from a lime mortar pointing specialist, but it will have to wait until next summer. (Not confident next door's builder knows what he's doing with lime mortar).
Am I being ridiculous? A lot of houses around here seem to have been repointed with standard cement mix. Not all of them have spalling bricks. My joints seem to be lime mortar (soft crumbly yellow material) with a smear of brittle grey cement mortar over the top.

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The process is the same regardless of the type of mortar, though. If your neighbor's builder did a tidy job there's no reason he shouldn't do the same for yours, provided the right material.
The current cement pointing will carry on damaging the brickwork if left.
Personally I would leave the bricks as they are still doing the job they were put there for. Finding a match could prove to be difficult to impossible, but in your favour they are already a wide range of colours with some even put in the wrong way round.
Out of interest there is a name for that kind of brickwork style (known as a 'bond') it's known as common bond or known by some as the American bond.
This looks to me to be either a very weak sand/cement mortar mix (is sort of yellow/beige) or else lime (looks like the pictures above). Since it is the back, and given the small scale, if i knew the lime recipe I could try patching until I can get it fixed properly when other work is budgeted for in a year or so?
And would temperatures next week ( 10-15 degrees and dry-ish in day be ok to do this?
It's not that visible so I don't need to have a super neat finish , which is why I thought I might try fixing myself.
Help appreciated if possible.
If you buy a French brand (ask, but I forget right now) it comes as a beige colour which helps tone it down as a lot of limes are stark white.
Once you have your lime you'll need to go around your local builders merchants and stone quarries (if they do sand) and get some sand samples, have a play until you get the colour right. You'll note sand typically comes as grey, orange and yellow.
The mix for my house was the above lime with equal amounts of a grey river sand and a yellow crushed stone. I managed to match the colour perfectly (it was a yellowy colour) and it now sits dry and in harmony with it's surroundings unlike the damp ruined blot on the landscape it was before.
OR you can buy premixed coloured limes. They are much more expensive, but have the convenience of being able to choose a colour from a chart like paint.
You can lime point yourself, but the big thing with NHL is immediate aftercare. It needs to set slowly and be protected from freezing for a good couple of weeks. This is done by starting with a damp wall, then covering over afterwards with an old sheet, quilt whatever you have to hand. This is to protect it from freeze and drying too quickly.
You can make a hot lime or quick lime, but that's Level 10 stuff and can be quite dangerous. I've read it's more forgiving to use, particularly in Winter as it sets differently. It was how they used to do it, NHLs are a more commonly used now as kind of modern equivalent.