Allardyce famously lasted just 67 days as England boss before resigning after being caught explaining how to get around FIFA regulations over transfers.
Southgate succeeded the 63-year-old and led England to the final four in Russia, only for Croatia to come from behind to deny them a place in the final.
"It's just a shame they didn't get to the final," Allardyce told 'Goals On Sunday'. They were 1-0 up with 20 minutes to go against Croatia and it didn't quite work out.
"You know me, I'd have got criticised for blocking it up and seeing the game out, because you're not supposed to do that now. France won the World Cup because they did that against Belgium [in the semi-finals].
"They saw what Belgium did against Brazil and even though (Didier) Deschamps was getting criticism from the French press for his style of play, they won it, which is what it's all about. Defensively they were so solid, and that's the basis for any success."
England next play in the UEFA Nations League next month and Allardyce believes that the pressure will be on the 'Three Lions' once again following their impressive World Cup showing.
"It will bring the pressure back on expectation because of the achievement. It was one of the youngest squads at the World Cup, so the experience will bode well but so many changes will happen between now and the European Championships.
"It's certainly changed from the one squad that I had and so many players have emerged, but will those players still be there?"