For English not being your first language, you certainly seem good at it to me…!
Try not to worry too much about conveying accents. I know you want to be authentic, but you don’t need to spell everything a character says out phonetically for the reader to understand where they’re from. In fact, I would argue that it’s better to avoid geet writin’ ya work oot like this yanarr because it can be very hard on the eyes after a while. If a reader finds something too difficult to understand, they might not feel compelled to read on. My advice is to be subtle; drop words in, abbreviate (i.e. runnin’, jumpin’, laughin’) and draw attention to the difference using character reactions and the like.
Watching videos to listen to the accents is good enough, although I would suggest going for ones where the accents are demonstrated, not imitated. Even if the video subject itself is not about the accent, just listening to the way someone speaks can give you a good understanding of how their accent varies from other types.
You’re right about accents in London being complicated though. Because London is vastly multi-cultural, there is a new type of accent around mostly used by the younger generation. Watching movies like Attack the Block will give you an example of it.
The typical London accent though that most people think about is that kind of posh southern one, not quite used by the Queen, but what you would learn via elocution lessons.
You need to have a think about what kind of accent you want to give to your London teacher, and make sure it rings true with where he lives and where he grew up.
Likewise with your older character from Durham. Pay attention to how a Durham accent differs from the broader Geordie accent using the resources below.
Best of luck and I hope this helps…!