He did well as the USA manager with arguably lesser players than England. He also got to a semi final in a world cup for Germany. Admittedly not considered a great success in that country.
Allardyce since leaving West ham. Bilic takes over getting better results and a more entertaining style of play.
I would go for Bilic rather than Allardyce, who has international experience as well. He comes across well as a pundit on TV as well.