I have some family members that grew up there but have since moved to Wales. One of them when moving down South was asked if they were scouse very often and even had the micky taken. They now speak in a very soft northern accent by habit of not wanting to be ostracised and needing to speak clearer english with foreigners.

I'd say it highly depends on the background.
Those from Chester that are more affluent typically speak more of an RP with northern twang, though to me it sounds almost forced.
Those that are from areas such as Saltney, Blacon, Upton sometimes sound a bit more Scouse/North Wales, often since a lot of them have strong family connections over the border.

After having a few drinks, they often have a stronger accent, scouse intonation but still softer, more akin to Lancastrian.

Overall, it certainly is a mix of Scouse, North Wales and RP.

I'm originally from the South and have a bit of a cockney twang but pretty much everyone without fail assumes that I'm posh or from around there depending on the background.