After calculating areas of the 1km grid shapefile rectangles, I can confirm that the single cells are not precisely 1km². The error is small, though. If your errors are larger, it is possible that your problem is related to a different issue (possible reprojection errors).
Small errors in the source dataset
Most of the calculated areas are less than 10m² off. (Mean value is 1km², standard deviation is 7m²; there are some outliers (corresponding to E-W-oriented lines in the far north): 210 out of 286000 rectangles have errors >100m², the largest errors being about 130m².)
Given the fact that on a 1km² rectangle, a 1mm increase in width propagates to a 1m² area increase, I am not sure if these shapefiles were designed to have a sub-meter precision. I have checked the coordinates of one of the most inaccurate grid cells: x error was 2 mm, y error 134 mm.
The error seems to be systematical, maybe due to inaccuracies during the initial generation of the grid coordinates: (See detail picture, blue/green shows "too small" areas, orange/red shows "too large" ones)
Although the error is pretty small, it seems unnecessary as grid cells are precisely defined. If you need more precise grid cells, either re-create them (inherit the naming scheme from the provided shapefile), or alter the coordinates of the individual shapes by rounding to the closest integer number. (This works because all errors are smaller than 0.5m, and linear unit is meter as the coordinates are given in OSGB 1936 projected coordinate system.) In fact, all Northing / Easting coordinates of the OSGB 1km grid should be multiples of 1000.)
However, with the grid lines being misaligned by a maximum of 13cm (less then 1cm for most grid lines), I would consider this a rather minor issue, which can most likely be neglected for many scenarios.
Possible coordinate transformation issue
Note that the British national grid uses the OSGB 1936 datum. This is based on the Airy 1860 ellipsoid. (See the OSNG wikipedia article for an overview information about datum shifts.)
In general, no transformation is without error, especially in the sub-meter scale. Do not perform transformations if you do not have to use them; for best precision, stay in original OSGB36 as long as possible and reproject in a final step, after calculations.
For reference, I projected the shapefile to PCS UTM30N (WGS84) and calculated areas afterwards. The error was much higher, most cells were about 140m² off from 1km².
Calculating areas after projection to GCS WGS84 produced an even larger error (approx. 650m² for most cells)
If you for some reason cannot use the original coordinate system for your project, eventually based on a different ellipsoid (e.g. UTM 30N with WGS84ellipsoid), check if coordinate transformations have been applied correctly.
For on-the-fly transformation, go to "Layer Properties" -> "coordinate system" -> "Transformations..." . Choose "GCS_OSGB_1936" in the "Convert from" box. The "Into" field should be your destination projection (e.g. GCS_WGS_84). Select one of the methods. (There is an overview article about transformations at ESRI blogs, the knowledge base article #21327 for a detailed list of transformations, and also a GIS.SE question that deals with transformations and OSGB).
For a permanent transformation of the shapefile, use "Arc Toolbox" -> "Data management Tools" -> "Projections and Transformations" -> "Feature" -> "Project".