Kdtyjb

I've just listened to the short BBC News audio podcast clip How British academics spied on the superpowers:

Tim O’Brien from the Jodrell Bank radio telescope discusses 50 year old recordings which show how Britain monitored Soviet space missions - and shared the findings with the Soviet Union.

He does indeed describe something more like assistance than spying (my attempt at transcription):

The Russians used to send Jodrell all the coordinates and the frequencies they were using, so that we could track these spacecraft, and of course we would then announce this to the world.

And I suppose in part, maybe, verify that they were really doing what they claimed to be doing, because when people would claim that they weren’t really technologically capable of doing this… because there was obviously a big political battle between these two superpowers.

So it had this sort-of role, even to the extent that Jodrell would track the spacecraft, and record the signals on to tape, and a Russian would fly into Ringway airport in Manchester as it was then, and someone from here would drive up… get off the plane and hand them a tape, and they’d head off to Moscow with that recording.

Was this purely scientific camaraderie (pardon the pun), or inter-government cooperation or something else? Clearly the Soviet Union's space program benefited from both data and PR; did the UK benefit as well?

SHORT ANSWER

Jodrell Bank's first 'coup', tracking Sputnik 1 in 1957 (without Soviet assistance), put it in the news and helped secure funding. It also led to a congratulatory telegram from the Soviets.

After doubts were expressed about Luna 1 (Jan 1959) being real, the Soviets sent the coordinates for Luna 2 (Sept 1959) to Jodrell Bank head Bernard Lovell to verify. This put Jodrell Bank in the news again.

Lovell, apparently both returning a favour and in expectation of future information which would again boost the prestige and importance of Jodrell Bank, sent the data recordings to the Soviets. Jodrell Bank subsequently received coordinates for Lunar 3 (which sent the first ever pictures of the far side of the moon) in Oct 1959 and again sent recorded data to Moscow.

After Luna 3, direct requests from the Soviets seem to have ended but transmission frequencies were published by TASS. Relations between Jodrell Bank and the Soviet Academy of Sciences became much less cordial upon the publication of the Luna 9 photographs in January 1966.

DETAILED ANSWER

The tracking of Sputnik (1957) put Jodrell Bank in the news and secured funding. This from the University of Manchester History Department blog:

The Jodrell Bank Observatory was catapulted to a position of global
prominence, Lovell’s critics were silenced, and new funds poured in –
not only from the British government, but also from the United States.

Bernard Lovell had

struggled to find the necessary resources to complete the telescope....Government funding for the project proved inadequate and
controversial....The project was heavily
criticized in Parliament as a boondoggle, and there were even calls to
have him arrested on charges of overspending public money, especially
since it was still unclear what purpose the telescope served.

Further, there was much conflict between him and the engineers. This provided another reason as:

At Jodrell Bank the satellite was used to show the public harmony of
engineers and astronomers, and the successful demonstration of working
telescopes.

Source: John Agar, Science and Spectacle: The Work of Jodrell Bank in Post-war British Culture

Agar further notes that:

The Soviet satellite was therefore appropriated differently by British
groups....For Jodrell Bank, Sputnik was used to demonstrate the
efficacy of their radio telescope. Why was this necessary?...to manage
internal divisions and contests of responsibility, and to counter
criticism of the construction of the telescope by the constitution of
public authority.

The fact that Jodrell Bank could track Sputnik was of huge interest to the US:

Desperate to monitor Soviet rocket launches, and to use Jodrell Bank’s
facilities to track their own rapidly expanding efforts, both the US
Air Force and NASA were soon paying substantial amounts for access to
the telescope.

Further, for a while it was the only telescope able to track an ICBM. Thus,

The Jodrell Bank telescope briefly became an “early warning device”
for a Soviet nuclear attack.

As you suggested in a comment, the providing of tapes to the Soviets was quite likely in acknowledgement of the Luna II (September 1959) coordinates received by Jodrell Bank. There was an exchange of information which was mutually beneficial, and perhaps a little professional courtesy also played a part as the Soviets

even sent him [Lovell] a telegram of thanks

following the Sputnik publicity.

This well-sourced Jodrell Bank Centre for Astrophysics (UOM) article notes that Levin was initially not overly enthusiastic in trying to track Luna II as they had failed to track Luna I (January 1959). However,

The duty controller at Jodrell Bank informed Sir Bernard that the
Russians 'had launched a rocket which would reach the moon on Sunday
evening'. A reporter phoned and asked what Jodrell Bank 'was going to
do about it'? To which Sir Bernard replied 'I am going to play
cricket'. Having been unable to track Luna 1 and receiving very little help from the Russians in locating that probe he was not so keen on wasting the observatory's time on another moon probe.

When Lovell returned to his office in the evening, "in his own words"

"And there, with the paper streaming
out on the floor, was a message from Moscow giving precise details
details of the frequencies of the transmitters in the Lunik and the
co-ordinates for the latitude and longitude of Jodrell bank giving the
time of lunar impact as 10 p.m. the following evening".

Further data from the Soviets followed the next day, and Jodrell bank was then able to confirm to the world

that an object made by humans had travelled from the earth to another
heavenly body!

Jodrell Bank's involvement in confirming this was not without controversy as some saw it as

'aiding and abetting the "Commies'".

Source: September 18 letter cited by John Agar, Science and Spectacle

Lovell's belief, though, was that

large radio telescopes could guarantee Britain a position of influence
between the USA and USSR

Source: Agar

Later, on the 1st of November,

a telegram was sent to AN USSR asking for "co-ordinates for the next
transmission from Luna 3", and the first batch of tapes of signals
from Luna 2 and 3 were sent to Prof. Nesmeyanov of AN USSR on 9
December 1959. In the accompanying letter Prof. Lovell thanked the
Academy for the data with pointing information for the Jodrell Bank
telescope. The final batch of Luna 3 tapes were sent to Akademi Nauk
on 14 January 1960.

No direct communication seems to have taken place between Jodrell Bank and the Soviets for subsequent luna (4 to 14) missions: instead, the Soviets published navigation fixes in TASS.

For the Soviets, Jodrell Bank helped them to prove to the world that their claims were true:

The Soviet Union also envisioned an important role for the Jodrell
Bank Observatory, which could provide independent verification of its
accomplishments. When the Soviet Union launched its Luna-1 rocket into
space in January 1959, many believed it had never happened. As a
result, when it launched the Luna-2 rocket in September 1959, Soviet
authorities sent Jodrell Bank the rocket’s coordinates and
frequencies; soon afterward, Jodrell Bank confirmed its existence.

Nonetheless, there were still doubters, even at Jodrell Bank; Lovell, though, was not one of them.