Setting CRS for comma delimited layer using QGIS

Question

I am using QGIS 3x.
I would like to overlay my .csv comma delimited layers in QGIS 3x properly. I have got the Ordnance Survey Maps as the raster players, which CRS is EPSG 27700 1936 British National Grid.
I overlaid these maps with my .csv data, where I set EPSG 4326: WGS 84. As a result, the data is visible, although there are some mismatches, as you can see below:

In the following example, you can see, that the difference is quite big, as the pole appears on the other side of the road effectively. I theoretically know, that the mismatch is driven by different CRS between the comma-delimited layers (WGS 84) and the map layer (British National Grid).
When I change the CRS for my comma-delimited layers, then they don't appear in my location anymore. Their new place is somewhat at the beginning of the coordinate system, as you can see below:

I would like to make sure, that all these layers match to each other as much as possible.
What should I do then?

Håvard Tveite · Accepted Answer

EPSG:4326 (WGS 1984, degrees of longitude and latitude) seems to be correct, as the deviations are very small. QGIS (and hence Proj) might have inaccurate transformation parameters for "EPSG 27700 1936 British National Grid" that could cause the effects that you are reporting.
But there could be other reasons - your positions may be inaccurate (could be the case if they are collected with a hand-held GNSS/GPS device), or there may not be enough decimals for the coordinates in your CSV file.
By the way, setting the CRS for a dataset will not transform it. You have to either Export-> Save as (and choose the CRS you want to transform to) or run the algorithm Reproject layer (from the Processing Toolbox).

Setting CRS for comma delimited layer using QGIS

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