It isn't the TSA you need to deal with, it is the British legal system that you will have to obey.
British laws do not allow for carry of locking knives longer than 3 inches unless you are camping or hunting in the appropriate locales, so you should limit yourself to SAKs and traditional slipcase knives just in case. -Do not think that you can pull a "I'm a dumb American!" defense, as foreign police and judges love to make examples of Americans. When in their country, obey their laws.
The knives ok for TSA carry on are fine for UK. TSA says 7cm maximum length. UK law says under 3" (7.54cm) non locking folder. I would leave the knife in your checked luggage untill you get to the UK, or take 2.
Spyderco UKPK is specifically designed to comply with UK knife carry laws. Generally, traditional (sub 3") slipjoints are fine, Tactical looking slippies and friction folders might be questioned (as the policeman may not be familiar with that kind of knife) but ideally should be ok, because they are S.139 exempt.
Please be aware that there are 2 situations in the UK regarding knives.
The law says that you do not need any reason to carry a non locking sub 3" folding pocketknife. They are exempt from all knife laws (apart from using it as an offensive weapon, where anything can become an offensive weapon by intent or actual use).
The general public (including many members of the police force) believe that all knives are banned because that's what newspapers (and rumour) state far too often. This means that a policeman who stops you and discovers the knife during a search, may believe he has you on a S.139 of Criminal Justice Act - Carrying a knife in a public place without "lawful reason" -
He may also believe there is no "lawful reason" for carrying any knife at all. He may arrest you or attempt to confiscate the knife. Know your rights and insist upon them. Do not accept a Police Caution (they may even try to pretend it's not a caution) This is a "Guilty plea" without the benefit of a trial. It will get you home early, without your knife and with a criminal record. If the Policeman arrests you, it's normally the custody sergeant who will be aware (hopefully) of the law and give you back your Section 139 exempt knife and send you on your way (no harm, no foul). Depending on which newpaper the custody sergeant reads, this might not happen. You may spend some time in a cell. During your PACE interview (Police And Criminal Evidence) state that you believe the knife to be exempt from S.139 of the Criminal Justice Act as it is a non-locking folding pocketknife with a blade length of less than 3". You therefore do not need to provide a reason for carrying the knife and you will not accept its confiscation and destruction. If ( for some ridiculous reason, it doesn't end there and) it still goes to court, insist on a crown court hearing (not a magistrates). Magistrates are unqualified upstanding members of the community, with little legal training and full of the usual NKP/ middle class prejudices (think New Yorkers) and many miscarriages/travesties of justice have been handed out by them over the centuries. Usually the CPS (Crown Prosecution Service, equivalent to the DA's office) realise the situation and drop the case early, but they are busy and may not see your case file until the day before 1st hearing.
Read some of the law section on Britishblades.com to become familiar with some of the issues. To compare reality, New York has stricter carry laws and any knife not specifically on the Offensive weapons list (flick knives, gravity knives, balisongs, shuriken, concealed/disguised knives, trench knives, knuckledusters, sword sticks etc.) are legal to carry (in a public place) provided you have good reason. You can do what you like on private property with the owner of the land's permission. This means that unlike Texas, UK subjects can own and carry bowie knives when hunting.
The biggest drawback of the british legal system is that without a constitution/bill of rights, justice is more dependant on circumstances than it is the "letter of the law". You sometimes have to go through a lot of inconvenience to uphold your rights. The thing is, you guys have had 237 years to balance constitutional rights with the dispensing of justice. Our constitutional rights come from the EU and were "imposed" on us in 1998. We're not used to the "freedom of dissent".