There is a misnomer in the question in that the beams are not cut to section at all, and not measured with any sort of ruler during manufacture. The sections are rolled on a hot rolling mill where billets of hot (hot enough to have softened) steel is driven between a series of rollers that force it to the required shape. See for example https://www.youtube.com/watch?v=6xnKmt_gsLs (particularly around 2:09)
(Other sections may be formed by cold rolling, but those to which the question relates - BS UB and UC sections - are hot rolled. Cold rolling is used for relatively thin sections, such as purlins or cladding rails supporting the cladding common on industrial sheds.)
The figures which designate a section are the 'serial size' (which is nominally in millimetres) and weight of the section. The dimensions of the sections are now explicitly defined by reference to SI dimensions. The current relevant standard is BS 4-1:2005 'Structural steel sections. Specification for hot-rolled sections'. http://shop.bsigroup.com/ProductDetail/?pid=000000000030128029
Section 3 of that standard states:
A section shall be designated by the serial size in millimetres (mm) and the mass per unit length in kilograms per metre (kg/m).
This form of designation shall be used in any enquiry and order.
A beam of 920.5 mm × 420.5 mm at 388 kg/m is designated a 914 × 419 × 388 beam.
However, the values of the units are not round numbers partly because they do originate as imperial values. BS 4 is (not surprisingly, given its very low number) one of the earliest British standards. The first version of BS 4 was BS 4:1903 http://shop.bsigroup.com/ProductDetail?pid=000000000030304821 In 1903, Britain was imperial (digression - BSI originates when Sir John Wolfe-Barry instigated the Council of the Institution of Civil Engineers to form a committee to consider standardising iron and steel sections on 22 January 1901, the day Queen Victoria died - about when Britain's empire was near its peak).
Sections very similar to BS 4:1903 sections are still in current use. Some of the nominal or serial size sections match exactly. For example, the BSB20 from BS 4:1903 was a 12" x 5" x 32 lbs/ft beam which had 0.55" thick flanges and 0.35" thick webs. In precise conversion, 12"=304.8mm, 5"=127.0mm, 32lbs/ft=47.631...kg/m, 0.55"=13.97mm, 0.35"=8.89mm. Now, 305 x 127 x 48 is a current section designation in BS 4-1:2005 (within table 2 in the standard) and has 14.0mm thick flanges and 9.0mm thick web.
(Although even where the nominal figures match exactly, the sections are not a precise match - BSB beams had tapered flanges, early UB sections had somewhat less tapered flanges, and modern UB beams have parallel flanges.)
However, if you actually put a ruler on this beam, it won't read as a round number in either imperial or SI units, because it isn't actually that size at all. A 305 x 127 x 48 UB is actually 310.4mm deep and 125.2mm wide. The designation is just a label, which reflects a nominal size termed the 'serial size'. The various beams designated as 305 deep vary between 303.8mm and 312.7mm deep.
This arises because you can roll all the beams from the same serial size from one set of rollers. For example, 305 x 127 x 48 UB, 305 x 127 x 42 UB, and 305 x 127 x 37 UB can all be rolled from the same rollers, you 'simply' set the rollers slightly further apart. You can see that from the section tables - those three sections are different depth and width, but the internal distance between flanges is the same (264.6mm) and the root radii are the same (8.9mm). The rollers can physically be the same set for all those sizes, the rollers that form the outer faces are set further away from the inner ones and the inner ones are themselves spaced further apart, making the flanges, web and also the overall dimensions thicker and greater.
That is, in summary:
The 'actual' dimension is not either of the designation dimension in millimetres or the conversion of that dimension into inches.
The actual dimensions are, however, defined in millimetres.
Some of those metric dimensions are the same dimension as a round number in inches, rounded to the nearest 0.1mm (but not all of them).
If you want to explore historical steel sizes, in addition to the BSs I've linked, there's a useful British Constructional Steelwork Association (BCSA) book - the Historical Structural Steelwork Handbook - http://www.steelconstruction.info/index.php?title=Special:ImagePage&t=Historical+Steelwork+Handbook.pdf