What test equipment do you have access to?
i.e. to measure Voltage & Continuity?
With suitable testers, it would take probably less than 15mins to verify if power is present..
if the thermostat is switching,
and if the cables are connected the correct way round..
Its generally best not to try guessing which wires are, Live, Switched live and Neutral..
when using devices that switch 230v
Is your picture the ONLY way you have tried connecting the thermostat?
Or did you try it another way first that blew the fuse?
If you did blow the fuse by wiring it incorrectly and putting a dead short across the thermostat terminals..
you could possibly have damaged the thermostat..
My first steps would be to disconnect the wires..
Then:-
[1] Verify that the thermostat contacts are still switching when you adjust the dial temperature up/down..
[2] Verify the supply polarity across the Brown/Black & grey wires..
[3] Verify that the boiler fires up with a temporary connection between the Live & Switched Live..
My only other thoughts would be...
Why were you removing the Nest thermostat in the first place..
And did it still work the boiler before you disconnected it?
OR.. Is there actually a boiler fault that you thought was the Nest..?
The photo you uploaded is only really useful if you also have an original photo as well to compare with.
Guinness