This question splits into a couple of parts, which I will answer one by one:
Where to keep the money:
This depends greatly on how long you think you will need to store the money.
If you are storing it for a short period (2-6 months) I would suggest just using a current account. You will not earn much interest but you are also unlikely to lose a lot. If you want to be extra safe split the money between several current accounts if you have them, to increase the amount insured by the government.
Alternatively some US banks also offer GBP denominated accounts, Schwab, with their global account (has no monthly fees or commissions and can hold USD, GBP and Euros) is one such account I know of but there are others.
Holding the money in a GBP denominated US account holds the very notable advantage that you are much less likely to need to file an FBAR or Form 8938 (Both declarations needed for the IRS when holding substantial funds abroad), but ill get into that later.
If you were planning on holding the money in GBP long term you could consider investing into a GBP denominated index fund such as a FTSE 100 fund, though I would be very concerned by this given the current state of the British economy. You can again either do this through a UK bank with brokerage features or a US bank with a GBP denominated investment account (I know Schwab also offers this in a Global account, I am sure many others do too.)
As I briefly mentioned above, you will almost certainly have to file an FBAR and Form 8938 to report the holding of substantial offshore funds, the penalties for failing to file these or doing them wrong are very high (hundreds of k $) so you may want to get the help of a lawyer or accountant to help with them. This assumes you keep the money in the UK for any time.
You will also have to pay tax on any interest or capital gains the money accrues while it is being stored (though fortunately you will probably not be double taxed due to the UK US tax treaty I will not go into here, allowing you to deduct some of the UK tax you pay on your US return).
Keeping the money in the US will simplify both of these things, regardless of if you hold it as USD or GBP, do not assume you have to convert the currency in order to move the money to the US. (Note for currency conversions over $100,000 many brokers offer very competitive fees and splits, meaning it may cost you less to move it to a US GBP account and convert it here.)
Regarding any Capital gains on the sale of the house. I believe these are not US taxable in most cases due to the UK US tax treaty, there is wording stating that physical investments tightly tied to foreign geographical land is usually not US taxed, but this depends on your circumstances, and given the amount of money involved the penalties for being wrong are very severe so get a lawyer or accountant to help if you are not totally confident, not random people online.
Do you really want to be a currency speculator?
How confident are you the GBP/USD rate will actually improve? Given the current situation I would consider it equally if not more likely it could further fall, at least in the next several years. You can certainly bet on this but I will be just that, a random gamble on how the rate will move. Do not consider it a surefire thing that you will recover any money, and may lose more by keeping the money in GBP. Remember, your expenses (seem to be) mostly in USD, so having a known quantity of USD is likely safer in terms you know how far the money will go, this is not a given in GBP.
If I were in your position, I would not want to invest into a known very shaky currency if I can avoid it, there are for sure, substantial opportunity to make money, but also a huge risk to lose all the proceeds from the home sale if things go south.
Think of this another way: If you owned your US property free and clear right now, would you take out a mortgage on it to try and speculate on the failing GBP? If not I would probably just convert the money to USD and put it towards your house. If you would follow whatever investments your risk tolerance allows.
(I would normally add a TLDR but this question is fundamentally very complex so reading this all is likely critical for the issue)
But to be honest with you given you are asking these questions it sounds like you desperately need the services of a qualified lawyer or accountant (or both) specializing in UK / US tax treaties who can both help you with these legal aspects and advise you on what to do with the funds, given the huge risks of crippling loss involved, both from penalties, as well as capital losses and currency risks.