If I was a betting man, I'd think that with what has gone on at Roma this season and from what I've seen with US business management, they'll give Dyche until the summer and make a decision then. Yes there is a chance of relegation in doing nothing, but there's that chance if they make a change also if whoever takes over has the same impact Juric had in replacing De Rossi. I personally think the chances of relegation are small regardless of who is in charge, and the three promoted teams will go straight back down again.
I'd expect maybe 1-2 signings (whether on-loan or permanent) this month), and that how they perform and are integrated into the squad will have a significant impact on whether Thelwell and Dyche are kept on or not. Often after a takeover, the existing staff get told the slate is being wiped clean, and that judgment will be based on how they do from that point on. I think thats very likely the case here. Selection of new players and the integration and performance of new signings is the key performance indicator, and doing poorly in that will ensure their exit.