For mute and moot to be homophones, the /j/ sound must be dropped in mute; this feature is called yod-dropping, and the presence of it is often dependent on the nature of the preceding consonant. Hughes et al. note that /j/ is dropped after consonants in much of eastern England, in regions like Norfolk, Suffolk, Essex, Cambridgeshire, Northamptonshire, Bedfordshire, Leicestershire, Lincolnshire and Nottinghamshire (2012: 69).
Homophony between these two words doesn’t seem to be prominent in any dialect of the UK, although we find a slightly higher rate of homophony between these two words in the North (16%) than we do in the South (7%).