Rather large amount of butterflies i would say.
If Britain did not attack,i could still see the deal that they offered the Crown prince come in play.. (stationing of British warships an if i remember correct 30.000 troops to help defend Jutland for Denmark "loaning" its navy to the British.)
Now the only way i see this deal being a reality is if Napoleon strikes first and invade from the south into Jutland, where the Danish Crown Prince and the army is waiting. Denmark seemed to have one competent general in Prince Frederik of Hesse. But let's be realistic and assume that France occupies the Jutland Peninsular.
due to there be no harsh winters for the Danish Straits in this period we won't see a repeat of the Swedish feat 150 years earlier, and France can't touch the valuable Danish Islands.
We now have a Denmark-Norway that is Hostile towards France and Allied with the British. Denmark in itself will never be able to field an army to go fight Napoleon, but it does open up for the british not having to fight Denmark in the Gunboat war.
Changes:
Norway won't starve.
Denmark will not loose Norway in 1814.
Denmark will not gain Lauenburg (as payment for Swedish Pommerania by the Prussians)
Prussia gains Lauenburg.
Denmark does not go broke in 1815.
Denmark still a signifigant naval power ( more spending on the navy later on, important for conflicts with Prussia later on)
Sweden does not gain Norway, What happens with Swedish Pommerania!?
What will Denmark's "payment" be for having the Jutland Peninsular occupied for 7 years. More of Germany is out of the question.. Abolishing the Treaty of Ribe from 1460 and integrating the Duchy of Slesvig-Holstein into the Kingdom of Denmark, would be a possibility. (Butterflying the war of 1864, the war of 1848 could very much still happen) French Caribbean islands another..
Would Frederick VI of Denmark accept the offer of the Swedish Throne in 1809 now that both nations are allied, and presumed friendly!?
So many interesting question and butterflies for Denmark if the British had not bombarded Copenhagen in 1807..