The question is 'Did Britain arm both sides'?
Both sides were sold arms etc. by industry but the British government were careful not to be seen selling government items to either side. Internally some government contractors were not above 'borrowing' items from the government production line for lucrative foreign sales and replacing them with later production.
There was nothing stopping British industry selling what they liked to whom they wanted to. The practice was that, if you had the funds, you could buy any weaponry that was for sale. How you got it across the Atlantic was your problem. Even by 1912 the purchase of rifles by the UVF was not illegal in itself (hence the Royal Navy could not seize them on the high seas.) It was the smuggling them ashore that was illegal. Equally the purchase of, say, enfield rifles was perfectly legal by either side in the UK as was shipping them out.
Arms dealers have always been used as intermediaries by governments. For example the US sold surplus guns to arms dealers when France needed guns for the 1870 war and they were purchased by the Defense Nationale. Many were captured by Germany and later sold to Turkey. Italian Veterlli rifles were sold to dealers who sold them to the UVF who smuggled then into Ireland and later gave them to the UK who eventually passed them on to Ethiopian allies in 1941.
What the British government did not do was sell any government arms to either side. Nor did they give any permissions for industry to sell as no permission was required.