I dunno. In the US, I’d say it wouldn’t go anywhere, as US law heavily favors the defendant (which would be BBC here) in a defamation suit, but English law has historically been exceptionally favorable to someone suing over defamation (here, Trump) than American law is, and this has actually led to some major disagreements in the past.

At one point, there were so many people trying to engage in venue shopping trying to find ways to sue over US works from the UK’s legal jurisdiction that the US passed a law saying that it wouldn’t recognize defamation lawsuits from courts that didn’t extend at least as much protection as the First Amendment did precisely for this reason, which was an exceptional situation.

https://en.wikipedia.org/wiki/English_defamation_law

English defamation law puts the burden of proof on the defendant, and does not require the plaintiff to prove falsehood. For that reason, it has been considered an impediment to free speech in much of the developed world. In many cases of libel tourism, plaintiffs sued in England to censor critical works when their home countries would reject the case outright. In the United States, the 2010 SPEECH Act makes foreign libel judgements unenforceable and unrecognisable by U.S. courts if they don’t comply with U.S. protections for freedom of speech and due process, which was made largely in response to the English laws.[3]

There were some revisions made to English law in response as well, so the situation has changed somewhat, but I don’t know to the degree:

The Defamation Act 2013 substantially changed English defamation law in recognition of these concerns, by narrowing the criteria for a successful claim, mandating evidence of actual or probable harm, and enhancing the scope of existing defences for website operators, public interest, and privileged publications. The 2013 law applies to causes of action occurring after its commencement on 1 January 2014.[4]

kagis

This article says that he probably can’t, just because he’s waited too long to do so:

https://www.newsweek.com/why-donald-trump-bbc-defamation-claim-doomed-attorneys-11033342

Mark Stephens CBE, a leading international media lawyer at Howard Kennedy LLP based in London, said that under British law, any attempt to sue in the United Kingdom is already out of time. “In the UK, his defamation claim is out of time. He had until midnight on October 27, 2025, to file, after which he can’t sue—it’s time-barred,” he said.

Britain’s one-year statute of limitations for libel cases would therefore prevent a domestic filing.

reads further

Oh, wait. He’s threatening to sue in Florida. Yeah, I don’t think that that’d succeed.

Trump’s lawyers have indicated that the case would be brought in Florida, where he resides.

In Florida, the statute of limitations for defamation (libel or slander) is two years from the date of first publication.

https://en.wikipedia.org/wiki/United_States_defamation_law

In 1964, however, the court issued an opinion in New York Times Co. v. Sullivan, 376 U.S. 254 (1964) dramatically changing the nature of libel law in the United States. In that case, the court determined that public officials could win a suit for libel only if they could demonstrate “actual malice” on the part of reporters or publishers. In that case, “actual malice” was defined as “knowledge that the information was false” or that it was published “with reckless disregard of whether it was false or not”. This decision was later extended to cover “public figures”, although the standard is still considerably lower in the case of private individuals.

https://en.wikipedia.org/wiki/New_York_Times_Co._v._Sullivan

New York Times Co. v. Sullivan, 376 U.S. 254 (1964), was a landmark U.S. Supreme Court decision that ruled the freedom of speech protections in the First Amendment to the U.S. Constitution limit the ability of a public official to sue for defamation.[1][2] The decision held that if a plaintiff in a defamation lawsuit is a public official or candidate for public office, then not only must they prove the normal elements of defamation—publication of a false defamatory statement to a third party—they must also prove that the statement was made with “actual malice”, meaning the defendant either knew the statement was false or recklessly disregarded whether it might be false.[2] New York Times Co. v. Sullivan is frequently ranked as one of the greatest Supreme Court decisions of the modern era.[3]

Trump would qualify as a public figure, so it’s really hard for him to win defamation suits under US law.

The Newsweek article I linked to also says that because the content in question wasn’t intentionally displayed in Florida, Florida courts probably wouldn’t have jurisdiction in the first place. But even if they did, I’d say that his chances of winning a case would be pretty slim.