Using OSGB36 will inevitably introduce further errors as it is an entirely outmoded co-ordinate system for anyone wanting highly accurate locations. Also depending on the conversion technique you used the RMS errors in conversion may well be in the order of 4-5 metres (i.e., if you do a straight conversion using proj4 utilities in ogr2ogr, postgis etc which I imagine is what JOSM uses).

So first of all you need to use the latest OSGB standard which I think is 2015. There’s an earlier one from 2002. For accurate conversion you need to apply specific corrections provided as a file by the OS. They have some detailed web pages about the conversions.

In general I would not advise trying to do such large scale rejigging as it’s pretty hard to get right: OSM objects will vary rarely be all created on the basis of the same underlying area. An accuracy of within 5m is what we can aspire to, trying to be more precise is not particularly worthwhile at present. In general when I’ve done this I’ve done it on smallish areas at a time and NOT by mass movement of nodes. You need to choose an area where the difference in position does not create obvious anomalies in surrounding areas which is what usually happens.

The StreetView base layer was created using the OSGB-2002 projection with the detailed correction, so this is a reasonable way to check for misalignment (mainly of larger roads, not buildings). The road network in OS Local Roads can also be used as a cross-check. At some point we ought to highlight roads which seriously diverge from this network (the techniques are not that complicated): there are many minor roads in Wales & the Welsh Borders which are significantly more than 10m away from true.