As a matter of fact, it's also a fairly broad question, because what exactly is meant by "war material"? The Great War was a gigantic conflict that required the British state to organise pretty much every aspect of life within the country and on behalf of its millions of combatants; in this sense, "war material" covered food and drink, clothing and the coverage of news as well as the manufacture of tanks and battleships.

With this said, however, it's possible to point out that the manufacture of weaponry, specifically, was regionalised to quite a considerable extent according to specialism. Shipbuilding, obviously, took place on the coast; there were many significant centres of construction in ports such as Glasgow (John Brown) and Liverpool-Birkenhead (Cammell Laird), and companies such as Thorneycroft, in Southampton, specialised in building small, fast ships such as destroyers, but by far the biggest two shipbuilding combines were Vickers, based in Barrow on the north-west coast of England, and Armstrongs, based in Newcastle-upon-Tyne on the north-east coast. Ships were also built in the Royal Dockyards at Plymouth, Portsmouth and Chatham, largely in order to give the government information about the true cost of the naval contracts they were signing with private shipbuilding concerns, but as part of a government attempt to improve the efficiency of the industry as a whole, these were increasingly focused on ship repair rather than new construction.

Steel manufacturing was ubiquitous, but there was a significant concentration of this industry in Sheffield. Birmingham was a noted centre of gun-making (BSA – which is famous now for motorbikes, but actually stands for "Birmingham Small Arms", was based there) and aircraft building was centred, at least at the beginning of the war, at the Royal Aircraft Establishment at Farnborough in Hampshire. In contrast, uniform-making took place largely in towns that had long specialised in cloth manufacturing, most of them in Lancashire.

London was of course a major industrial centre, and the largest of its establishments was Woolwich Arsenal, which a few years earlier had lent its name to the famous football team founded there, and which was was a very important site for explosives manufacturing – though heavy guns (such as guns for battleships) were built there too. But here, too, efforts had been made to diversify production, keep costs down, and improve reserve capacity. So, for example, the government had backed the creation of the Coventry Ordnance Works, a joint venture of John Brown, Fairfield and Cammell Laird, in the years before the war by guaranteeing contracts to the new venture.

With regard to statistics, the armaments industry was covered in absolutely exhaustive detail by the British official history of the war – I would direct you to the 12 volumes comprising the History of the Ministry of Munitions in that series, which contain statistics sufficient to keep any historian busy for a lifetime.

While /u/mikedash has given a good overview of British industry in general, I would like to do a more specific dive into the British shipbuilding industry. This goes beyond the basics of slipways and fitting-out basins, to the works where engines, armour and armaments were produced. Factories that produced equipment and ammunition for the Royal Navy were spread across much of the country, and as such, they give a good idea of how distributed British manufacturing was.

We'll start out with shipyards, without which the other industries were unimportant. The two main centres for British warship building were the Tyne in north-east England, and the Clyde in western Scotland. Tyneside, centred on the city of Newcastle, had four major shipyards: Armstrong's works at Elswick and High Walker, Palmer's at Jarrow, and Swan, Hunter and Wigham Richardson at Wallsend. Clydeside, around Glasgow, also had four big shipyards - Beardmore's, John Brown, Scotts', and Fairfield - plus large complexes of smaller yards at Old Kilpatrick, Dumbarton and Port Glasgow. These included the major Yarrow's shipyard, which built a large number of the RN's destroyers. There were also important shipyards elsewhere in the country. Harland and Wolff, at Belfast in Northern Ireland, were a major mercantile shipyard, and built a number of smaller warships. Cammell Laird at Birkenhead built a similar mix. Vickers' yard at Barrow was mainly focused on military shipbuilding, especially submarines. The south coast had the three Royal Dockyards at Plymouth, Portsmouth and Chatham. Plymouth and Portsmouth were major centres of battleship production, while Chatham built smaller ships. There were also smaller yards, like Thorneycroft's at Southampton. The Thames Iron Works in East London constructed a number of major ships in the 1890s and early 1900s, including the dreadnought Thunderer, but closed down in 1912. This was a fairly common fate for London yards, which had high labour costs, had to pay more to move materials south from the industrial centres of the North, and high land costs making it hard to expand.

Warships could only be built on the coast, or on major rivers like the Tyne, Mersey and Clyde. But the typical warship needed engines, guns, armour and a whole bevy of smaller items. These could be built inland, as long as there were good transportation links to the shipyards elsewhere.

Armour plate was built by five major companies. All five companies were directly associated with shipyards, being owned by the same companies. Armstrong Whitworth had a major armour plant at Openshaw in Manchester. Vickers' shipyard at Barrow had an extensive armour works, as well as the armour works in Sheffield where the company had started out. In addition to Vickers', Sheffield had two major armour conglomerates; John Brown and Cammell Laird. Finally, Beardmore's produced armour at their Parkhead factory in Glasgow. Warships also needed large quantities of structural steel, rivets and bolts, made from mild steel. These were not produced by the armour manufacturers, but were instead ordered from ordinary steelworks. These could be found around the country, but those that focused on shipbuilding steel were largely found around the major shipbuilding centres - Teesside in the Northwest and Motherwell near Glasgow were major steel centres associated with the shipbuilding trade.

The armaments companies were similarly largely owned by shipbuilding companies. Armstrong Whitworth had armaments factories at Openshaw (Manchester) and Elswick (Newcastle). Vickers had three armaments factories - one producing heavy guns in Sheffield, one for smaller guns at Erith, and one for machine guns in Crayford, both in Kent. Cammell Laird, Fairfield and John Brown both had stakes in the Coventry Ordnance Works, mostly based in the eponymous town, but also with a works at Scotstoun (near Glasgow) for gun mountings. Beardmore's Parkhead works produced guns as well as armour plate. There was also a nationalised gun factory, the Royal Gun Factory at Woolwich. This was part of the larger Royal Arsenal complex there which produced guns and ammunition for the Army. Ammunition was produced by a similar mix of public and private companies; propellants were largely produced by public factories, while projectiles were mainly produced by private industry. Until the start of the war, about a third of British cordite propellant was produced by the Royal Gunpowder Factory at Waltham Abbey, just north of London. The other two-thirds was produced by a mix of companies based around the country - examples include Nobel's at Ardeer on the Clyde, the National Explosives Co. of Cornwall, and the Chilworth Gunpowder Co. in Surrey. The start of the war imposed vast demands for cordite, leading the Navy to establish its own Royal Naval Cordite Facility. This opened in 1915 at Holton Heath near Poole, and supplied most of the RN's cordite needs. Shells were mainly produced in Sheffield by Hadfield Steel Foundry Co Ltd, Thomas Firth & Sons and Vickers. Vickers also produced some shells at Barrow and Erith, while Armstrong made theirs at Scotswood in Newcastle. Some smaller calibre shells were produced at Woolwich Arsenal, which also filled shells produced by private industry.From 1916, the Royal Ordnance Factory at Cardonald, near Glasgow, began to produce heavy shells; this, along with Hadfield, soon became the main producers of these. Torpedoes were produced by the Royal Naval Torpedo Factory at Greenock and the Whitehead Torpedo Works in Weymouth (and frmo 1915, Caton in Lancashire).

The British marine engine industry had, in the Victorian era, been fairly widespread, with some fifteen engineering companies spread around the country. They were largely based on the major rivers - the Thames, the Clyde, and Tyne. However, the advent of the steam turbine forced the Thames companies out of the warship business. They already faced high costs for labour and material, and could not cope with the large investments needed to produce large steam turbines. This resulted in a concentration of engine-building around Tyneside and Clydeside. In the Newcastle area, the main engine-building companies were Parsons, Wallsend Slipway and Hawthorne Leslie. John Brown in Glasgow were highly successful, having licensed the American Curtis turbine design, modified it to produce the often used Brown-Curtis design. As well as producing them in their own factories, they sub-licensed the design out to a number of companies - Fairfield and Yarrow in Glasgow and Thorneycroft in Southampton. Other shipyards also had their own engine-building facilities; Vickers, Fairfield, Scotts, Beardmores and Harland & Wolff all had their own facilities.

Along with the turbines, warships needed propellers, gears and shafts to make them move, and a whole host of auxiliary machinery to run pumps, fans and so on. Propeller shafts were largely produced by the major steel contractors discussed above, such as Firths of Sheffield or Armstrong-Whitworth in Manchester. Gearing was produced both by the engine manufacturers and by a number of specialist companies, including David Brown of Huddersfield. Propellers were built by two main companies, both based in London. The Manganese Bronze and Brass Co were based in Millwall while J Stone & Co had factories in Deptford, and from 1916, Charlton. Auxiliary machinery covered a wide range of electrical and mechanical systems; as such, there were a vast number of contractors. These were spread all over the country. Electrical equipment was produced in London, Manchester, Norwich, Loughborough, Rugby and Glasgow. Diesel engines were manufactured in Lincoln, Grantham, Basingstoke and Glasgow, while auxilary steam engines and air compressors were made in Birmingham, Bedford, Peterborough, London and Glasgow. The two most important companies were Weir of Glasgow and Allen of Bedford.

Finally, we come to more miscellaneous topics. Fire control equipment was one important area. Rangefinders (and periscopes for submarines) were produced by Barr and stround in Glasgow. The first mechanical fire control computer, the Dumaresq, was made by Elliott Brothers in Lewisham in London, while Vickers produced its range clock at Erith. Elliot Brothers also produced the more advanced Dreyer Table, while its competition in the Pollen Argo Clock was made by Cooke & Sons of York. Refrigeration systems for food and to cool the magazines where propellant were stored were produced by the Pulsometer Engineering Co of London and Reading. Compasses, indicators and other gauges were produced by a number of companees - Elliott Brothers, Evershed & Vignoles of London, Heath & Co of London, Chadburn's of Lancashire and the Cambridge Scientific Instrument Co, amongst others. Anchors, chains and other fittings were produced by a number of contractors, often in Sheffield or the West Midlands.