Cornetto

Short Answer: Road movie

This series is only barely one. Shaun of the Dead, Hot Fuzz, and World's End share no characters or plotlines, only a production team and a sense of humor. Yet, here we are.

They, respectively, parody zombie movies, buddy-cop movies, and sci-fi movies. So the logical thing is to keep going until they work through all the major genres.

So far, they have taken place in the present. So they could go for spy, superhero, kaiju, prison, road, or Die Hard-style movies (which have become their own subgenre. All of these are ripe for the type of knowing winks and physical violence that mark the Cornetto series, named for a British ice cream brand that appears in all the movies.

OK, so they share a production team, a sense of humor and... Good Humor. Or is that "Humour."

In any case, the necessity of including ice cream rules out some of the other action movie types they could skewer, like fantasy, Western, and war movies. 

I guess there could be ice cream in a war movie. Ice cream has been around almost as long as war, and this brand has been around since 1959. So that gives us all the wars since then to work with. Yay..?

Kaiju and superhero movies must be pricey with those special effects. Plus superhero parodies abound. Making fun of prisons is probably not cool anymore. 

Which narrows it to spy, road, and Die Hard. Again, there are as many Bond parodies as there are Bond movies. And the days of school and mass shootings make a Die Hard situation less funny.

So a road movie it is:

A guy has to move across England. He talks his friend into going with him and helping him unload and unpack at the far end: "I'll pay for the flight back."

He says they'll make it fun. They will stop in cool towns to sightsee and party. It'll be one last glorious road trip. The friend agrees.

Cue the Cornetto sense of humor and... action! References! Fights! Explosions! Gore! More references! etc. etc. 

They finally make it to the new place, with very little of his possessions intact, only to find more... plot to deal with. The promised roommate (or just mate) taps out, so the friend-- who wasn't doing anything anyway-- decides to stay as be the roommate.