Definitely India. They plan to launch a man to orbit no earlier than next year.
Followed, perhaps, by Italy/ESA. It's uncertain after that point, I would say Japan, Iran, or depending on the fate of RocketLab, NZ.
It is unlikely anyone could get a man into space within a year without an already-functioning space transport system. India is about 14 months away, assuming no further schedule slips. It takes more than 6 months for existing space capsules to be prepared for launch - see Inspiration 4. (And that was SpaceX!)
The bulk of this post is looking into Launch vehicles. A space capsule is at least as hard as the rocket itself, so this is only half the picture. However, some countries have experience with spacecraft but not as much with rockets; particularly Italy/ESA - UK & France/ESA - Canada, in roughly that order. If you were considering multinational teams, there might be some good pairings. But I don't keep track of space capsules as closely as launch vehicles.
Countries not on your list of exclusions, with orbital-class rockets, who have looked into indigenous manned launch capabilities:
India: Has an unusually successful space program, and are actively working on a manned space capsule. They are planning an initial manned launch of Gaganyaan next year.
Europe: Have a very well-established space program, but have recently been moving very slowly. Have looked into manned spaceflight in the past (Hermes in particular), and are looking into it again, but no real progress toward it as of yet (SUISE and Nyx are two to keep an eye on). Adapting SpaceRider's 600kg payload for manned flights - on Vega (Italian) - is likely the 'quickest' option, but a solution launched on Ariane 6 (French) is a better long-term solution.
Japan: A high-quality space program, but hampered by their semi-artificially limited launch numbers. Have looked into manned space systems in the past, but would take some time to bring one into operation if needed. An iteration on the Fuji capsule would be a good start (design for simplicity and low cost).
Iran: Of their surprisingly diverse array of orbital rockets, only Simorgh SLV has the payload to launch a person - barely. However, the only manned spacecraft proposal light enough to be lofted to orbit with its ~300kg payload is MOOSE. Interestingly though, unlike the Kiwis and Israelis who have similar rockets, the Iranian space agency has advertised that it could launch an astronaut on this rocket. It also has some heavier rockets in development (Soroush) which should be able to launch a more respectable manned vehicle, but their introduction date is uncertain.
Countries not on your list of exclusions, with orbital rockets, who have not (publicly) looked into manned launch capability:
New Zealand: While Rocket Lab is now a US-based company, the Electron was designed and is mostly built in NZ. If the entirety of the USA were to spontaneously lose all ontological inertia, NZ would likely still be able to launch Electron. It is a small rocket though and while technically capable of carrying a manned reentry vehicle (again, see MOOSE), it isn't ideal. Not to mention no studies have been done on this. Rocket Lab has investigated a manned space capsule for Neutron, but that rocket is being built and launched exclusively in the US (Darn ITAR), so it couldn't easily be construed as an NZ launch.
Israel: Shavit-2 launches rarely, but has been around for many decades. It, too, could only launch a MOOSE-class manned reentry vehicle. Also, all Israeli launches necessarily end up in a very retrograde orbit.
N Korea: has a few different types of orbital rocket, but only its most recent and most powerful, Chollima 1 (which has yet to reach orbit) has enough payload for manned flights - and at 300kgs, only barely (yes, MOOSE again).
S Korea: has not had many rockets, but last year succeeded in its first indigenous rocket launch. Nuri has a payload of 1.5t - enough to launch a Mercury capsule with margin to spare.
Others: There are several countries that, if pushed, might be able to develop an orbital launch capability in a reasonable timeframe. The first that spring to mind are the UK - it used to have an orbital rocket, but cancelled it; and Pakistan - they have plenty of experience in missile weapons systems, not too unrelated from rockets. Of these two, I would put the UK ahead in getting a man to space due to its more focussed experience (SSTL, Orbex/Skyrora, UKSA, REL/Skylon, BIS, etc). But at this point, the timescale would be around a decade, not months or years.
After writing this answer, I do feel it is a better fit for worldbuilding.se. Perhaps a migration is in order?
