It is rather difficult to predict anything with Paul but if the plot fails and it is found that Britain is behind it, one can imagine his feelings. On the top of these feelings there would be something else: just at that time Nelson with his squadron sailed into the Baltic with the expressed goal to destroy Swedish and Russian navies. Judging by his correspondence, he was quite certain in his ability to “defeat” Paul. Initially, situation favored him because Russian Baltic fleet was split between Revel and Kronstadt. However, while he was doing whatever he was doing, the Revel squadron sailed for Kronstadt. In OTL the whole affair ended with Paul’s assassination but in your AH Paul is alive and Nelson is sailing toward Kronstadt.
No matter how (un)successful would be Nelson’s further actions (goog luck with attack on Kronstadt fortifications), the public sentiment toward Britain in Russia would swing from a neutral to negative at such an offense. Needless to say that even in the most lucky scenario Nelson would not be able to cause any serious damage to Russian Empire because it’s power was on the land (and good buy to the British ships in the Russian ports, British property in the empire, etc.). The diplomatic relations would be broken and effect could be lasting for years. Nappy and the French are firmly established as the good guys and Perfidious Albion is ... well .. perfidious.
No 3rd Coalition and the rest is open to the speculations.