There is no provision for an ex-military Sea King to fly public transport anywhere, let alone (especially) in the UK.
These aircraft were built solely for military operation, with the exception of one Sikorsky S61A that was delivered to a commercial operator as an FAA restricted category helicopter and was unfortunately destroyed in an accident last year. The S61A was a designation for an FAA Restricted Category type certificate, as a basis of an acceptable design standard (it's a long story), but it also appears as the model for some foreign military deliveries including Malaysia and Denmark. All the original aircraft Serial Numbers listed on this TC were removed at the last revision.
The H-3's in the desert will probably come out at some time, but are of equally limited application. They are only eligible for certification under Restricted Category, and are no longer eligible for certification under the original Sikorsky FAA Type Certificate H2EA. The machines in the desert are parked in full military configuration, including folding heads, etc.
There are two means open to you to certify ex US military aircraft by the FAA.
The only way they have been certified to date, is under FAA Restricted Category which will require you to obtain your own Restricted Category Certificate approved by the FAA. The limitations of restricted category include no carriage of passengers (amongst other things). FAA restricted category regulations require that the aircraft be previously operated by the US military - any other user at any time disqualifies the aircraft from eligibility. A further Catch 22, is that under the most recent certification change, you have to be in possession of at least one aircraft that will be eligible for certification, PRIOR to making application for a Type Certificate!
The second option would be to undertake a complete transport category certification programme and conform the aircraft to the appropriate regulatory requirements. This would include all systems and components - a major undertaking! There is absolutely nothing to stop you from conforming an ex-military aircraft from any source, if you have the ability to conform it to an approved Type Certificate. This has been the means of certification of a vast array of aircraft, both helicopter and fixed-wing over the years. Of course, in the past many aircraft were essentially the same between commercial and military models. As you can probably understand the scope and expense of this undertaking, it is easy to understand why no-one has done it (or probably ever will).
The other option to find something similar is the "short" S61N or S61L, where the original commercial aircraft is modified to remove the airframe plug that created them, and restore them to the same dimension as the Sea-King. Agusta built a couple of S61N1 "Silver" which was essentially a commercial build short S61 but they were not exactly a commercial success.
I see that some of the Royal Navy Sea-Kings are being retired now, maybe you could get one just to fly around in one on a permit and be the envy of the Gazelle boys?