The original base plate you show is reasonably genetic, it can be used for CH (central heating) and DHW (Domestic hot water) or just one. I assume only used for CH? the British regulations require an earth wire for any fixed unit, but you can have a flex to class II item without an earth, but using the earth wires as a conductor is rather bad practice even if not against the regulations.
Nest e is designed to work either on/off (digital) or up/down (analogue) in this case analogue is far better than digital, so if the boiler is OpenTherm enabled then far better to use that option, but depends on the boiler as to if it has option or if it can be wired without removing gas tight panels.
However since using an earth wire for control it would seem not fitted by a professional in first place so would not be worrying about using a gas safe person to any great extent, but would do alterations just before next service so it can be tested.
If that wiring was found on an EICR it is possible it could be given a code C2, personally I would not award C2 not even sure if would give it a C3, but some inspectors are rather pedantic, giving C2 for items which were never wrong when fitted, like plastic CU and no RCD protection, so if you need an EICR doing then would be better if this is corrected.
