United Kingdom

‘within three weeks’

If so, it seems pretty unlikely to me that the people negotiating can be doing much in terms of modifying things from the pre-tariff situation, and Trump is likely to do what he did with NAFTA->USMCA

change very little, and then spend time giving the impression to supporters that he's drastically modified the trade environment (Fox News: "Trump has solved our trade problems that Biden permitted to happen with the best trade deal ever"). I mean, trying to complete any kind of meaningful free trade agreement tends to take far longer than that.

https://www.piie.com/blogs/trade-and-investment-policy-watch/how-long-does-it-take-conclude-trade-agreement-us

Table 1 Duration of US free trade agreement negotiations (in months)

US FTA partner	From launch date to signing	From launch date to implementation
Jordan	4	18
Dominican Republic	6	37
Bahrain	7	30
Oman	10	45
Korea	13	69
Australia	14	22
Israel	15	29
Morocco	16	35
Costa Rica	18	71
El Salvador	18	37
Guatemala	18	40
Honduras	18	38
Mexico	18	31
Nicaragua	18	38
Canada	20	32
Peru	23	56
Singapore	29	37
Chile	30	36
Colombia	31	96
Panama	38	102
Average	18	45

On top of the fact that this would be off-the-charts short for a meaningful FTA in any case, neither of the two "shortening" conditions that were found exist here; it is not a US election year, and while the UK is nominally a monarchy, the monarch holds no power and Parliament is, no doubt, going to be involved in any substantial change in trading relationship.

Despite the small sample, two variables are significant in explaining the delay between launch and signing.

1. A king. Having a monarch reduces the length of negotiation by about half. Only four agreements took less than a year, and three were with Bahrain, Jordan, and Oman. A king surely has more leeway to carry out reforms he deems reasonable. (The fourth was the Dominican Republic’s negotiation to join the Central American Free Trade Agreement or CAFTA, though it benefited from joining late, which may suggest that late entrants to an already negotiated TPP could also face shorter delays.)
2. An election year. Agreements that are signed in a US presidential election year end up taking about 40 percent less time than agreements signed in other years. This makes sense: Negotiating presidents want to close agreements that they started, which will be part of their legacy. The urge to close is real: More than half of the US agreements were signed in election years and of course the TPP, if implemented, will add to that group.

In the UK's case, there was some prior discussion about a UK-USA FTA, so maybe they could bootstrap off that to reduce the negotiation time, but I have a hard time believing that even an administration-friendly, Republican-majority Congress is going to sign off on whatever the Trump administration negotiates in a major FTA without having some kind of input.

https://en.wikipedia.org/wiki/United_Kingdom%E2%80%93United_States_Free_Trade_Agreement