There is a position of Deputy Prime Minister, and also First Secretary of State, sometimes both are used sometimes neither, sometimes the same person holds both titles, sometimes they are combined with meaningful cabinet portfolios there have been long periods when neither is in use.

They are essentially honorary titles usually used by the PM to confer special status on a senior colleague to either reward them for something or to stop them from causing too much trouble. Occasionally they are titles given with specific additional responsbilities (e.g. Atlee was Deputy PM to Churchill during WW2 and broadly speaking looked after domestic affairs while Churchill focussed on the war effort).

There's no order of precendence like in the US, and there's no hard and fast rule on who takes over in the event of death or sudden incapacity of the PM, officially it is the decision of the monarch, but he/she will only appoint a PM if they are sure that they have enough support to form a government.

If there is no official deputy PM then presumably the cabinet would meet and agree a figurehead/spokesperson until it became clear who amongst them had enough support to form a government at which point the monarch would ask that person to form a government and become PM.