Many UK regional newspapers, if not most of them, took a political stance in their reporting at the time of the ABW (the Labour Party was only formed on 27th February 1900). This encouraged towns with a good population to have two papers - one with Conservative views, the other with Liberal - and the Conservative papers were fully in support of the war, whilst Liberal papers were less so.
The first report I came across of Adam Lawson's death was in the Lancashire Daily Post, published in Preston; its politics aren't known, but it included war news from all over Mid and North Lancashire, so it's not surprising that Lawson's death was reported.
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DEATH OF AN ACCRINGTON SOLDIER.
....The death from enteric fever is reported from Bloemfontein of Private Adam Lawson, of the Coldstream Guards, who was the son of Mr. James Lawson, of 10, Monk-street, Accrington. Deceased was 23 years of age.
Lancashire Daily Post, Saturday 26th May 1900
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I next looked for Lawson in the Accrington Observer - nothing, other than a death notice inserted and paid for by his parents.
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DEATHS.
LAWSON.—At Bloemfontein, Private Adam Lawson, of the Coldstream Guards, the beloved son of James and Ann Lawson, No. 10, Monk-street, Accrington. Died of enteric fever in his 24th year. Deeply regretted.
The Accrington Observer, Saturday 26th May 1900
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Finally, I looked in the Accrington Gazette, which proclaimed itself to be the only Conservative newspaper in the Division.
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DEATH OF AN ACCRINGTON SOLDIER AT BLOEMFONTEIN.
....It is with deep regret that we record the death, at Bloemfontein, of Private Adam Lawson, of the 1st Battalion Coldstream Guards, which occurred a week yesterday (Friday), from enteric fever. Pte. Lawson's parents reside at 10, Monk-street, Accrington, but he was born at Oswaldtwistle. He was 24 years of age. Lawson joined the Coldstream Guards on the 19th of May, 1897, prior to which he was a plate moulder at Messrs. Howard and Bullough's Globe Works. He has been at the seat of war for three months but during that time his parents have only heard from him once—when he landed at Capetown. He was with Lord Roberts's Army, and it is evident that he was in the bulk of the fighting prior to the occupation of Bloemfontein. We sympathise with his parents in their sad bereavement.
The Accrington Division Gazette, Saturday 26th May 1900
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From the above it's safe to assume that the Accrington Observer was a Liberal newspaper, but whatever its political leaning, it seems odd not to report on the death of one of its fellow townsmen in South Africa, fairly important news for Accrington.
