Let's say I am going to visit country A before visiting country B. Can I apply for the visa of country A before applying for the visa of country B because of time constraints? In a particular case, when i plan to travel to UK then US, can i apply for a UK visa before applying for a USA visa? can this be a reason for refusal of UK visa because of the lack of proof to leave UK considering itinerary (possibility of a refusal to enter USA)?
Yes and yes.
You can apply in any order you wish. The appications will be evaluated on their merits.
It will be easier to convince the UK that you will leave again if you have a visa for the US. It will be more difficult if you have no US visa yet. This increased difficulty could mean that an application will be refused that would have been granted if the US visa had been present. We can't predict the outcome in a specific case or the probabilities.