I can't answer specifically your question because there are a lot of 'if's involved. It's almost like the stock market these days. You'd be exchanging on speculation. And I don't know about the Post Office changing that much. I would doubt, but a phone call will tell you.
What I would be watchful for is the currency exchange rate. It's mad right now. A couple months ago $1.70 = £1. I just had to change some money over and the rate was $1.97 = £1. (The UK bank charged me £28 to do that, BTW.) I understand it's expected to soon go to something like $2.10 = £1.
If in your shoes, I would probably make the exchange as soon as you can before the dollar gets any weaker. Who knows when it's going to bounce back to "normal"?
You might want to take a look at what this website has for information:
Foreign Currency DirectThey may not be useful for you specifically, but their web site does contain information on how to transfer money when emigrating from the US to the UK, including conversion rates.
I don't know this company to recommend, just something I found online.