Interesting. I don't think it would really escalate into a full blown war simply because neither side would want it, the Brazilians certainly don't want to face one of the great powers of the day and the British aren't going to want to because it'll cost money for little to no gain. As far as I'm aware they were already able to trade freely in Brazil unlike with China where they used the Opium Wars to force them to open up to trade. I'm sure that there might be a certain amount of jingoism by the public back in the UK but the government would probably want it settled as quickly as possible with as little cost so that commerce, the main interest of the Empire, was allowed to start flowing again.
This. IOTL basically there was no war because neither country wanted it. The problem was caused by two incidents. First, in June 1861, the merchant ship Prince of Wales grounded on the coast of Rio Grande do Sul, near the border with Uruguay. The crew abbandoned the ship and went to the port of Rio Grande in order to ask for help. But while they were there the local population thought that it was a shipwreck, and they looted the cargo. When the crew arrived back to the ship they noticed the lack of the cargo, and sent a letter complaining about this to the British ambassador in Rio de Janeiro, William Dougal Christie, who asked in "non polite" terms to the Emperor Pedro II an indenization. The Emperor refused.
Secondly, in 1862 two drunk sailors of the Royal Navy, members of the crew of the HMS Emerald, were dressed as civilians in Rio and got involved in a street fight with Brazilian sailors. What sparked the fight was a competition involving women (probably prostitutes). The police arrested the British, but released them the following morning.
However, two days later, the Brazilian Minister of Foreign Affairs sent a letter to Christie, asking to let the two British sailor be suffer penalties according to the Brazilian Justice. At the time, Brazil and the UK had a legal agreement that stated that British citizen could only be judged by the Bristish courts. This angered Christie, who again demanded the indenization for the loss of the Prince of Wales, the arrest of the Brazilian sailors, the firing from the Police of the Brazilian policemen who arrested the British sailors and a formal excuse from the Empire to the UK. If these demands were not accepted, they he threatened to block the port of Rio and to capture Brazilian merchant ships.
Of course the government refused, and in December 1862 Britain
did blockade the harbour of Rio de Janeiro, and they captured 5 Brazilian ships. Then Brazil asked for international arbitration, choosing Leopold I of Belgium to decide the matter. Pedro II also asked for an indenization for the Brazilian ships captured, and demanded a formal excuse from London, that refused. Only then we cut our diplomatica relations with Britain, in May 1863.
After that, Pedro II decided to pay the indenization asked by the British, as he believed that the matter was more related to offended honour of the country than only money. But Leopold I decided the question in favour of Brazil, and then the Emperor asked the money and the formal excuses from the British ambassador. He didn't receive any of them. Only in 1865, one year after the beggining of the Tripple Alliance War, Britain replaced the ambassador and sent formal excuses to the Brazilian government.
So, it was never a casus belli. The British already had a lot of advantages in Brazil, and we couldn't lose our greatest market. It was all a matter of national pride. However, if there a possibility of conflict is higher, then probably the government wouldn't risk to make an intervention in the Uruguayan civil war in 1864. Without the Brazilian invasion Paraguay would have no reason to declare war against Brazil, and so we would have no Tripple Alliance War, or at least a delayed one. However, it would only happen while Mitre was in power in Argentina, as he had the same aims for the region as the Liberal government in Brazil. If the Conservatives gain the Parliament or Mitre and his allies are not in power then the cooperation between Brazil and Argentina doesn't exist, and the whole context of the war is changed.