How does the lockdown affect the application and admission processes in the Western Europe, Scandinavia, and the US?

I work at a Finnish university. My department established around April a moratorium that new hires could only begin their contract starting August 1st. There are also recommendations to avoid receiving visitors (and I guess new hires, but I'm not sure since I haven't attempted to hire anyone since earlier in the year) until the Autumn. We also have the policy that you need to be in Finland by the start of your contract (i.e., you can't start remotely), meaning that travel restrictions from specific countries could prevent you from getting hired in practice. Things are at a stall during July because of the holidays, but as new outbreaks take place throughout Europe and the situation continues to be quite bad in some countries, I expect the university to update their official policies in early August, after admin people are back from the holidays. Generally, even though Finland had an easier time than most of Europe during the pandemic (so far), the university has been quite strict about enforcing remote work, and we're preferentially working remotely until mid August, when we will start gradually to go back to normal. But this can be revisited at any time.

This is for my particular university in Finland. I believe other Finnish universities have similar policies. Other Nordic countries have probably similar rules and practices, although I don't know about Sweden, where the Government has been notoriously bad about their handling of the pandemic, compared to the other Nordics.

In summary, the situation is quite volatile and things can change rapidly, although right now looks much better recruitment wise than a couple of months ago.

The main problem for you now is not COVID-19 but simply the summer holidays. If you are looking now, you won't find much. The number of job offers may increase again in September, October.

I cannot answer this for all of Western Europe, Scandinavia, and the US. Policies may differ between continents, countries, universities, and perhaps even departments.

At my university, which is in a Western European country, there are currently 13 job vacancies for postdoctoral researchers. These are all specific positions in specific departments, so whether any job opening is suitable for you, I cannot answer. If you find a vacancy that is suitable for you, the starting date at my university can be whatever you and your supervisor will agree on. The supervisor may be bound by restrictions imposed by the funding that provided the postdoctoral vacancy.