Japan would be the hardest hit, it has little resources on the home islands, the population loss would be crippling (along with every other country)
Japan gains its power through its manufacturing and financial sectors, food is commonly shipped in, same with resources for manufacturing. Nuclear war basically shuts down trade between countries, even if Japan recovered there is not enough resources to "bounce back" without exerting control over overseas territory. The smaller territory of the home islands means more radioactive areas, and considering the usable land is a fraction of the total land, most of Japan would be uninhabitable
I doubt the remnants of the 7th Fleet would stick around, they would return to the US mainland or head to Australia depending on what the orders from the US Government, they would not be willing to be used as a "big stick" to scare other countries
Moreover
To say any country would "recover well" is academic, most countries as a whole would be shattered into city states, any remnant of the Government would have to make themselves legitement, through territory or conquest
I.e. the US government would likely only control a small portion of the Continental US, most territory would be split between independant states (New California Republic from fallout as an example). Overseas American Assets would try at best to return to the US to help with recovery and all that fun stuff, troops and equipment will be returned to the US (if available)
Just imagine the USS Nimitz powering a city, thats likely what most of the pre war navy is going to be used for
With the war, infrustructure and logistics are gone, 70 percent of people in 83 would not know how to gather food (hunting and such), with nearly 90 percent of the global population destroyed survivors will have to basically start at the stone age with the help of pre war tech. To get anywhere near 83 technology will take a long ass time
The Southern hemisphere will be hit less then the north, but technology will be devestated by the EMP effects, and given the IRL poorer state of the Southern hemisphere (besides certain countries like Australia) they will take decades to recover to just a pre war state, to say it will be unscathed is silly, as global trade would cause alot of issues when surviving technology breaks down and needs spare parts built in the US or other NUKED countries
I.e. Wikipedia wouldn't exist because any surviving technolgy like the internet would be for government and military use only, plus the brain drain of scientists dying and moving to other occupations (surviving) would cause alot of the "existing knowledge" to be gone. technological progress would grind to a halt
World War Z goes into this better then me
The biggest issue would be the global satellite network being hit hard by EMP effects, global communications would be back to radio and relays, the South has no experience building rockets (in 83 only launcher was black arrow, built in UK), likely it would take beyond 2050 to launch a satellite
Also to, this Timelines Democrats leaning more left with no republicans is a fallacy and takes IRL changes without modifying them for a nuclear war, the Democrats would be hardline leaders, any semblance of OTL US politics would be none-existant
not to mention having an election would be a pain in the ass, its likely the presidents will have lifetime limites for the near future