Operation Paperclip was just one component of a concerted effort by all of the Allied powers to capture and exploit technologies developed by the German military-industrial machine, particularly in aviation (where the V-2 guided missile and Me 262 jet fighter represented major technological advancements.)

The British certainly conducted these kinds of knowledge-transfer operations, but on a markedly smaller scale than the Americans as Soviets — largely as a result of institutional anxiety about security, and generally tending more towards short-term secondments rather than permanent relocation.

In 1944, the British and American general staffs created the Combined Intelligence Objectives Subcommittee, a joint working group to coordinate the Allies' efforts to locate and capture scientific, technical and industrial intelligence during the liberation of Europe. After D-Day, SHAEF G-2 (the intelligence department of the Supreme Headquarters Allied Expeditionary Force) organised 'T-Forces' — mobile formations which would act as the operational arm of CIOS to secure key scientific and industrial sites and personnel.

All of the Allied powers recognised that to replicate and expand on German research (without having to painstakingly reverse-engineer), they would also need insight into the R&D process — which meant talking to the people directly involved.

Here's the official report on Operation Backfire, a British field test of captured V-2 rockets conducted in Germany in October 1945. The author, Maj. Gen. A. M. Cameron, provides a pretty detailed case study on how the T-Forces worked, what was done with the information they acquired, and how German personnel were used.

Section 3 in particular gives some insight into the thinking behind Paperclip and similar operations (emphasis mine):

After the German A-4 Division surrendered at the end of April, 1945, 107 of its officers and men had been selected by M.I. for interrogation. They were chosen from amongst those with the longest practical experience and the most knowledge of improvements and simplifications in launching methods. They included officers who held important operational, administrative and technical appointments in the Division. At the end of May, the party was taken over for operation "BACKFIRE."

[...]

As soon as it became apparent that the building of rockets was going to be a complicated and tricky business, it was decided to supplement this party of soldiers by civilian scientists and other technicians. Permission was therefore obtained to select 79 technicians from amongst those whom the United States had concentrated at GARMISCH PARTENKIRCHEN, and they arrived at CUXHAVEN at the end of July.

24. It was realised from the beginning that the handling of the Germans would be a delicate matter. The problem was very different from normal interrogation. If the difficulties of building rockets were to be overcome, and the launching procedure learnt, active co-operation rather than passive obedience was essential. At the same time there was uncertainty as to how far their apparent willingness to co-operate could be trusted.

[...]

25. In order to keep the Germans in a co-operative frame of mind it was thought desirable to ameliorate their lot to some extent. Permission was therefore obtained to increase their pay and their rations, and towards the end of the operation to pay a bonus, pro rata according to responsibility and technical efficiency.

That same rationale applies across these extraction operations: German technical expertise could be invaluable to future R&D, and their active cooperation was essential. There were short- and long-term considerations at play here: some German scientists and technicians simply had useful information they could share; others were specialists whose skillsets could be valuable on a more permanent basis.

Operation Matchbox, which you referenced, certainly existed — and it absolutely involved the relocation of German scientists and engineers to the UK and its colonies (likely as a reward for information-sharing, and to keep them out of the hands of the Soviets.) But its scope and execution are rather poorly documented — the National Archives in London apparently have a bunch of Ministry of Aviation and Ministry of Defence documents on Matchbox, but they haven't been digitised.

From a paper on the history of the Defence Research Policy Committee (which oversaw British military research in this period):

Operation Matchbox actively removed German scientists and technicians from Soviet zones of influence. The extraction and transfer of information was always as important as the exploitation of skills. Over 400 German scientists were interrogated at the Beltane Schools in Wimbledon by 1946, whereas only 30 of these were offered jobs. From 1948, having "skimmed the cream", the flow of scientists and information slowed and reversed: many Germans wished to return home, their presence often caused unspecified "morale problems" in British laboratories, and the strategic decision had been made to "employ German scientists in Germany who might otherwise be tempted to work for the Russians, it was necessary to build up healthy German science."

From the report of the Deschênes Commission, a 1987 Canadian public inquiry into war criminals in Canada:

In the course of time, the Commission was apprised of operation "Matchbox" aimed at securing for the Allies the greatest number of top-notch German scientists. The operation presented a serious security problem inasmuch as those scientists might well have been involved in the Nazi war effort. Fresh in everybody's mind must then have been the discussions which would eventually lead to the trial at Nurnberg of the I.G. Farben directors.

[...]

The security screenings were conducted by two distinct panels established by the British Board of Trade. The DARWIN panel was assigned the responsibility of recruiting and screening of scientists for industry while the other panel, under the Deputy Chiefs of Staff Committee, dealt with scientists destined for military research. In both situations the panel was to verify if the applicant was a member of the Nazi Party and whether he was sufficiently qualified to contribute something worthwhile in his field of endeavour in the host country.

This paper on Operation Surgeon, a similar British operation, offers more insight into the how and why of British scientific recruitment efforts:

By November 1946, the British Scientific Staff had produced records on 500 German aeronautical experts and selected 74 for inclusion in lists for long-term employment in the UK. Scientists from the MoS (Munitions) exploiting rocket technology at Trauen were also instrumental in selecting German missile specialists who contributed to the overall total of 122 scientists in which that branch of the [Ministry of Aircraft Production] had an interest.

Between July 1945 and November 1946, the British Scientific Staff interrogated an array of high-grade German aeronautical experts and made their recommendations under the provisions of the [Deputy Chiefs of Staff Committee] Scheme. Over the same period, acute barriers hindered the translations of these recommendations into a large-scale recruitment programme. The original Defence Committee endorsement for the DCOS Scheme had explicitly restricted German recruitment ‘to that minimum who have a real contribution to make’. This policy to limit recruitment reflected an accumulation of security and political considerations.

In part, the strict recruitment limits reflected a perceived security dilemma: how to extract the maximum amount of knowledge the German experts could provide for the post-war military research and development (R&D) programme while at the same time reducing ‘to the absolute minimum the knowledge they [could] acquire of our own work and intentions’. The aspiration of the DCOS Scheme was to ‘to bring German scientists to the UK in order that their knowledge might be exploited and after we had obtained what we wanted from them, ... they would be sent back to Germany’