The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

ALTER TABLE `newer` ADD INDEX `newer_idx_thoroughfare_building_name` (`THOROUGHFARE`,`BUILDING_NAME`);
ALTER TABLE `upcdata` ADD INDEX `upcdata_idx_building_name` (`Building Name`);

The optimized query:

SELECT
        t1.`Geo ID`,
        t1.`Sub Unit Geo ID`,
        t1.`Sub Unit SW ID`,
        t1.`Building No`,
        t1.`Building Name`,
        t1.Road,
        t2.ID `Matching NEW_ID`,
        t2.Sub,
        t2.SUB_BUILDING_NAME,
        t2.BUILDING_NAME,
        t2.BUILDING_NUMBER,
        t2.THOROUGHFARE,
        t2.E - t1.Easting `East Difference`,
        t2.N - t1.Northing `North Difference` 
    FROM
        upcdata t1 
    JOIN
        newer t2 
            ON (
                t2.E * 1000
            ) BETWEEN t1.Easting - 25000 AND t1.Easting + 25000 
            AND (
                t2.N * 1000
            ) BETWEEN t1.Northing - 25000 AND t1.Northing + 25000 
            AND t1.Road = t2.THOROUGHFARE 
            AND t1.`Building Name` = t2.BUILDING_NAME 
            AND t1.`Building Name` <> '' 
    ORDER BY
        t1.`Geo ID`