The highest team total in Pakistan vs England test matches is 708 allout, by Pakistan, made at The Oval back in August 1987..
Traditionally a batting beauty, Kennington Oval hosted this last test match on Pakistan's tour of England and visitors won a good toss which nearly sealed the series for them as they had 1-0 lead coming into this match. Opener Rameez Raja and No.3 Manzoor Akhtar missed out but Mudassar Nazar (73), Javed Miandad (260), Saleem Mallik (102) cashed in. Captain Imran Khan nailed in a century late on day two to push visitors in commanding position before they were allout on day 3, breaking their highest ever total against any country, beating 674 for 4 against India.
At that time, it was 6th biggest total by any team and 2nd largest against England. Ian Botham gave away 216 runs from 52 overs he bold, the maximum by any English bowler at that time. But, it couldn't win the game for Pakistan.
Pakistan have gone past 600 run mark versus England 4 times.
Highest total by England in a test match against Pakistan is 558 for 6 wkts declared at Trent Bridge, which they made in reply to 157 allout by visitors and went on to win the match by innings and 129 runs. Denis Compton scored a double century (278) while opener Reg Simpson made a hundred.
Here's a list of totals in Pak v Eng tests, limited to scores larger than 500.