Extradition from the United Kingdom, under the Extradition Act 2003, normally happens in the context of an treaty between the UK and the other country, coupled with an official designation by the UK government of that country as either a "Category 1" or a "Category 2" territory. If there isn't such an agreement, ad-hoc arrangements are still possible for a specific proposed extradition. Section 194 of the Act allows the government to deem that place to be a Category 2 territory (possibly with some modifications to the rules) for the purposes of that request only.
This recently happened for Japan, which wanted to extradite three people from the UK for alleged robbery of a jewellery shop in Tokyo in 2015. This is apparently the first time Japan has proposed such an extradition, and there is no treaty. Instead, the UK government concluded a "memorandum of cooperation" with Japan and issued the certificate required by Section 194. The three people challenged the legality of the UK government's actions in a judicial review, but lost; R (Chappell and Others) v SSHD  EWHC 3281 (Admin).
For an alleged crime which took place outside the UK, it's also possible in some circumstances for an arrest and prosecution to happen in the UK. That is another potential avenue, aside from extradition, where another country passes information to the UK police and hopes that they will proceed.
British law would not allow somebody to be arrested and handed over, other than by the process laid down in statute. It is a basic principle that the state is not allowed to detain people outside of the legal structure. Aside from such statutes as the Habeas Corpus Act 1679, which enacted a general prohibition on the sending of prisoners "into any Parts Garrisons Islands or Places beyond the Seas" and which only another statute could supersede, modern human rights law would not permit this sort of action. In a reverse example, where somebody was abducted from South Africa and brought to the UK outside of a formal extradition process, the House of Lords found that the abduction was such a grave abuse of power that the prosecution could not be allowed to proceed (R v Horseferry Road Magistrates' Court, ex parte Bennett  1 AC 42).