I'm not sure the U.K. is a very good example. We now have this system with the EU that allows for extradition requests that many EU members face, and because they have this principle of legality, they don't exercise any discretion. For most countries where this principle doesn't exist, there is a decision taken as to whether an offence is serious enough to justify the resources needed in order to make an extradition request, but within the EU, that doesn't really apply. We face requests that range from what you would see as very serious to those that some people would consider to be much more trivial.
I would add one caveat, which is that it is quite hard to categorize seriousness unless you understand the context within which these crimes are being prosecuted. What may seem trivial to you in a particular country can be an endemic problem. If there's a low standard of living, the theft of a chicken can actually be a relatively serious offence, even though in a different country, that might seem pretty trivial.