As Mamyalynne's link shows, the rules have been changed but possibly not by quite as much as you seem to think, JB9.
For example, Basil D'Oliviera, Tony Greig, Allan Lamb, Andrew Strauss, Matt Prior, Jonathan Trott, Craig Kieswetter and Kevin Pietersen were all born in South Africa and played for England well before the current rules came into effect. (Of the current crop, Jason Roy, Keaton Jennings and Tom Curran were born in South Africa too).
Colin Cowdrey, Bob Woolmer, Robin Jackman and Nasser Hussain were born in India.
Devon Malcolm was born in Jamaica.
Andy Caddick was born in New Zealand (as Ben Stokes was).
Phillip DeFreitas was born in Dominica.
Phil Edmonds was born in Zambia.
Graeme Hick was born in Zimbabwe.
Mike Denness was a Scot, who played for both Scotland and England.
Roland Butcher was born in Barbados (as Jofra Archer was too).
Derek Pringle was born in Kenya.
Since the 'England' team actually represents the England and Wales Cricket Board, it should be no surprise that there have been Welshmen (such as Pat Pocock and Simon Jones) playing for England in the past.