Surprisingly enough, in England, screens for identity parades did not come in at all until the late 1980s.
A poke around the British Newspaper Archive (paywall, sorry) found this article from Nov 1986, talking about the success of the £4,000 Merseyside Police had spent on "one of Britain's first two-way mirror systems" (presumably what we would call a one-way mirror), and this article from July 1987 where West Mercia Police would be "among the first forces in the country to introduce one-way vision screens" following "an experiment in which the screens were installed in some stations earlier this year". It's not quite clear when they became universal, but it suggests forces were starting to roll them out across their regions at that point.
There seem to have been other experiments at the same time; this July 1986 article notes the success of an experiment in secretly videoing a suspect who had refused to take part in an identity parade, and circulating that along with video of nine other men doing the same thing.
So, why had it changed? Following some high-profile mistaken identification cases, the Devlin Committee on Identification in Criminal Cases in 1976 looked into all forms of identification of suspects. They specifically considered one-way screens (para 5.55), which had been recommended by an earlier report but did not feel that "the problem of the nervous witness is grave enough to require this solution".
(The earlier report that recommended screens was the Thomson Committee on Criminal Procedure in Scotland, 1975, Cmnd 6218 (in para 12.07); I haven't yet tracked down a copy of that to see what exactly it said, but the Devlin report suggests that they recommended something very much along the lines of the modern one-way screen system. It's unclear if these approaches were indeed adopted in Scotland prior to being adopted in England & Wales)
The Devlin report recommended giving the guidelines for identification parades statutory force, which happened with the Police and Criminal Evidence Act 1984. This gave the Home Secretary powers to issue guidelines, and it is these new regulations which will presumably have established using screens, first as an experiment and then as a universal rule. This review outlines some post-Devlin changes, but focuses on video identification (authorised 1991, common from 2002, now pretty much ubiquitous) and does not say anything about why screens were adopted. I can't find anything specifically discussing it in Hansard, which is odd - it's the sort of thing you would expect to be announced there.
(A note on terms: identity or identification parade is the most common form in the UK, though the US "line-up" does also get used; they're the same thing. "Parade" is presumably from the military sense of assembling and mustering troops for a particular purpose - "sick parade", "pay parade", "church parade" - rather than implying a public event.)