Tbh since I've been alive there hasn't been many great British Managers (bar Sir Alex) so my list will probably be terrible.
1. Sir Alex Ferguson - For obvious reasons.
2. Sam Allardyce - He may not play the most exciting Football but he has done well with multiple clubs, especially Bolton Wanderers, and he has only been relegated once in his whole career which is very impressive.
3. Harry Redknapp - He did extremely well with Portsmouth, and he took Tottenham from a team that was bottom of the table and was a complete basket case and turned them into a Quarter-finalist in the Champions League in one of his years. Admittedly he could've done better with us, but his record is still more solid than most of the British Managers that I've seen.
4. Martin O'Neill - He had Leicester, Villa, Wycombe, and Celtic all doing quite well for themselves, in particular he had Villa playing the best Football that I've ever seen them play personally, and even the one year challenged for the top 4.
5. Steve Bruce - He did extremely well with us having us last four consecutive seasons in the Premier League and even when we got relegated he got us straight back up the following season. And he did quite well at Wigan, Hull, and imo he has made Villa look respectable again after the carnage they have gone through since O'Neill's departure from the club.
An honourable mention to McLeish for his accomplishments with us, Rangers, and the Scottish national team.