I'm going to bump this since I think it's a really good POD.
It's a bit confusing reading about Paul's switching sides from being in an anti-Napoleonic alliance with the British to allying with Napoleon. This section from Wikipedia helps:
"Although by the fall of 1799 the Russo-Austrian alliance had more or less fallen apart, Paul still cooperated willingly with the British. Together, they planned to invade the Netherlands, and through that country attack France proper. Unlike Austria, neither country had any secret territorial ambitions, they both simply sought the removal of Bonaparte.
[32] The campaign started well, with an English victory in the north, but when the Russian army arrived in September, the allies found themselves faced with bad weather, poor coordination, and unexpectedly fierce resistance from the Dutch and the French, and their success evaporated.
[33] As the month wore on, the weather worsened and the allies suffered more and more losses, eventually signing an armistice in October 1799.
[34] The Russians suffered three-quarters of allied losses and the English left their troops on an island in the Channel after the retreat, as England did not want them on the main land.
[35] This defeat and subsequent maltreating of Russian troops strained Russo-English relations, but it a definite break did not occur until later.
[36] The reasons for this break are less clear and simple than those of the split with Austria, but there several key events occurred over the winter of 1799-1800 that helped: Bonaparte released 7,000 captive Russian troops that the English had refused to pay the ransom for; Paul grew closer to the Scandinavian countries of Denmark and Sweden, whose claim to neutral shipping rights offended the British; Paul had the English ambassador in St. Petersburg recalled and England did not replace him, with no clear reason given as to why; and the English, needing to choose between their two allies, chose the Austrians, who had certainly committed to fighting Napoleon to the end.
[37] Finally, two events occurred in rapid succession that destroyed the alliance completely: first, in July 1800, the British seized a Danish frigate, prompting Paul to close the English trading factories in St. Petersburg as well as impound British ships and cargo; second, even though the allies resolved this crisis, Paul could not forgive the English for
Admiral Nelson’s refusal to return Malta to the Order of St. John, and therefore to Paul, when the English captured it from the French in September 1800.
[38] Paul’s drastic response was to seize all English vessels in Russian ports, send their crews to detention camps and take English traders hostage until he received satisfaction.
[39] Over the next winter, he went further, using his new
Armed Neutrality coalition with Sweden, Denmark and Prussia to prepare the Baltic against possible British attack, prevent the British from searching neutral merchant vessels, and freeze all British trade in Northern Europe.
[40] As Bonaparte had already closed all of Western and Southern Europe to British trade, England, which relied heavily upon imports (especially for timber, naval products, and grain) was seriously threatened by Paul’s move and reacted fast.
[41] In March 1801, the English sent a fleet to Denmark, bombarding Copenhagen and forcing them to surrender in the beginning of April.
[42] This fleet then prepared to head to St. Petersburg, but by this time the conspiracy had already assassinated Paul and Alexander made peace shortly after taking the throne.
[43]"
TL;DR Essentially, Anglo-Russian relations broke down over British interference with neutral shipping. Russia joined the
Second League of Armed Neutrality, which Britian considered a form of alliance with France. This lead to the attack on the Danish fleet at Copenhagen, and would have continued (had Paul not been assassinated) to attack the Russian fleet in St. Petersburg. Since our POD is Paul not being assassinated, the attack on the Russian fleet would have gone ahead and firmly placed Russia in Napoleon's camp.