This is probably true, strictly according to the wording of the question, but the Blades were not the first football club to have United in their name. That honour is claimed by the successors of Hanover United, formed in 1875 in Chiswick, West London.
Hanover United later belonged to the Regent Street Polytechnic – a philanthropic organisation that aimed to give working class men in London "somewhere to learn and reap the physical and character–building benefits of exercise". It was founded in 1882, as the Young Men's Christian Institute, by Quintin Hogg, an Old Etonian and the great–grandfather of the second Baron Hailsham – a prominent Conservative politician in the second half of the twentieth century (who served as Lord Chancellor under Margaret Thatcher).
Hogg's polytechnic used the premises that had been vacated by the Royal Polytechnic Institution, which was founded in 1837 but closed in 1881. It was later part of the Polytechnic of Central London, which is now the University of Westminster.
Hanover United FC seems to have been founded independently of the Polytechnic (in the last days of the Royal Polytechnic Institution), but by 1892, when it became a founding member of the Southern Football Alliance, it was known as Polytechnic FC – the name that it still bears today. (The club's first opponents in the Alliance were Tottenham Hotspur – who won 2–1.) Polytechnic FC currently plays in the Southern Amateur League, of which it has been champions for the last three seasons.
Sheffield United FC was named after a cricket club, which was formed in 1854 by the merger of several local cricket clubs and claims to be the first sports club or association in England to bear the word 'United' in its name. The cricket club was based at Bramhall Lane; The Wednesday FC had used its ground from 1881 to 1887, but left following a dispute over rent. In 1889, the ground hosted an FA Cup semi–final between Preston North End and West Bromwich Albion (which Preston won 1–0). The crowd of 22,688 paid £574 to watch this match, and the cricket club, encouraged by the potential revenue that football was evidently capable of generating, decided to form its own football club.
So, although Sheffield United was not the first football club to include the word 'United' in its name, it may well have been the second, and it's almost certainly been using the word for longer than any other. This would mean that it can claim today to be "the oldest 'United' in English football."
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