So if the Navy wants a new top of the line capital ship how would it go about getting it? What kind of approval and funding process would be needed and how long would it take? How long would it take to go from design to construction? Also if costs can be somewhat equalized for inflation, what would the differences be?
I guess I'm really wondering about the differences between time periods.
So this is quite a broad question OP. Do you mean particularly about the specifics about launching a ship to commissioning it? If so, this history.navy.mil piece on christening, launching, and commissioning a ship in the Navy will give you historical background on the specifics there from the origins of the Navy to today.
If you're talking about the politics of ordering such a project, that's a little easier to understand. In reality, the process hasn't changed significantly at all since the founding of the nation!
Note that the Secretary of the Navy (SECNAV) has been a position that has existed 1798 and was a cabinet level position from 1798 until 1949, when the SECNAV and Secretaries of the Army and Air Force were made subordinate to the Secretary of Defense (SECDEF).
Ever since the Naval Act of 1794 - which authorized the construction of the original six frigates of the United States (side not: one of them, USS Constitution, is still commissioned!) - the template has been the same: Congress approves the executive branch's request for warships and stipulates the rules for funding.
Note that even in the Naval Act of 1794, Congress put stipulations on size of ships, guns carried, and even rations! Once approved, those warships are built according to agreed upon specifications between Congress and what the Navy/President desires and negotiated with the shipbuilder as well.
Navy.mil: Federal/Quasi-War & Historical Overview of the Federalist Navy, 1787-1801
As thus, the process is very political. It should be noted that warships have a lot of tradition around this. Even referencing a ship as being commissioned is a reference to the fact that the ship beings to the nation/head of state (hence USS United States Ship and HMS is Her Majesty's Ship) and is authorized/funded by the legislative branch for the executive branch to utilize (another reason why ships are always commanded by officers, who hold office granted by the President/head of state). It's also a note that many capital ships (battleships, for instance) are named after states and political figures (aircraft carriers today, for instance) to show both their importance to the nation and because its a way to gain political support, but I digress.
It should be noted that because of this, the size of the Navy and the urgency with which ships are built/designed/upgraded varies significantly with the political winds of the time.
For instance, under Buchanan, with a strongly isolationist foreign policy, the US Navy just prior to the civil war was relatively small but expanded rapidly as the Union Navy enacted a strategy of blockade on the South and requested a corresponding increase in ship size/design.
Fast forward to the WW2 era, and the process hasn't changed. In 1940, the "Two Ocean" Navy Act was a bill by US Senator Carl Vinson (which the nuclear powered aircraft carrier USS Carl Vinson is named after), crafted by Chief of Naval Operations ADM Stark, which called for a massive spending increase ($4 billion in 1940, $68 billion in 2016 dollars) to create a truly two ocean navy in consideration for the looming hostilities. This increased was staggering - they expected over 18 new aircraft carriers, 7 new battleships, and a slew of new cruisers/destroyers/submarines.
Today, the process remains the same. Each year, the Secretary of the Navy and the Chief of Naval Operations and their engineers/planners work in cooperation with shipbuilders to design new ships that are included in the annual DOD Budget Request link here for 2016, which is then approved by the SECDEF and President and sent to Congress for authorization/funding.
The process remains highly political (see: the controversy around the Littoral Combat Ship in Congress) but that is the one constant.
As to your other questions about cost/time to build, that's a complex question as costs/construction methods vary considerably by era. A wooden-hulled warship of the Buchanan area wouldn't take much longer than any other wooden-hulled ships of the era would. The Civil War itself would spur construction to higher speeds, while the post-war drawdowns would slow or even see ships get cancelled.
It should be noted that by the Kennedy era and later, warships have become so technologically advanced that mass production like in WW2 is virtually impossible. The US's modern supercarrier, for instance, can only be built in Newport News (by Huntington Ingalls) as it is the only place in the US with the space/size to build such massive ships. In fact, Huntington Ingalls is the sole designer, builder, and refueler of the US supercarrier fleet. The USS Gerald R. Ford was laid down in 2009 and launched in 2013 and will be commissioned in six months time, which should give you a good idea for how long a major capital ship today can take. The HMS Queen Elizabeth for the Royal Navy is taking even longer - laid down in 2009, launched in 2014, and not to be commissioned until mid 2017.