If Ukraine resists for many weeks, why wouldn't Putin use a nuclear weapon against them? [...] I don't see how Putin has much to lose with this strategy (except that it might horrify his direct subordinates or population enough for them to oust him), and it's a sure-fire way to win Ukraine.
Yes, the worst thing you can do to someone who has nuclear weapons is to put them in a situation where they have nothing left to lose. I'm sure a major reason for the sanctions on Putin-associates is to destabilise his allies sufficiently for them to want to remove him. It's difficult to imagine him backing down voluntarily or due to Western pressure.
As for him winning Ukraine, there's a huge difference between taking control and holding control. He can certainly capture the key cities and assume command of the country, but the longer this goes on and the bloodier it gets, the more the Ukranians unite against the common foe. I read
an interesting article by Yuval Noah Harari (author of the excellent 'Sapiens') the other day, where he argues that by effectively uniting Ukraine against him, Putin is ensuring that he can never retain control of the country in the longer-term.