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The pound's going to take more of a beating, and any Brits who are aware of Bitcoin will start thinking about buying some. If they do well out of it they will tell their friends who aren't aware of Bitcoin. Also if the euro starts taking a beating the same could happen all over Europe. That demand could push Bitcoin over $1000 before the Autumn even without the approaching halving.
I doubt the pound's going to take much more beating. Dollars and Euros are being printed at an alarming rate, and both can't really stop doing that, only try to not print faster than anybody else. The UK books are more in balance than the EU books, therefore the UK has the option to slow it's money press a little. I don't think the pound will have a large effect on BTC.
But there are other interesting things: UK citizens may face higher transaction cost to/from Europe, and especially small consumers will get more trouble with import taxes, i.e: a maximum amount they can spend tax-free each year.
(European consumers can now spend a relatively small amount in the US. Pass that limit, and substantial taxes (~20%) are added.)
UK citizens may also spend more in other countries in the world, now they are similarly priced to products for Europe (especially from China)
More worldwide payments that need to be cheap and fast for small payments, is good for bitcoin.
I think these 3 have more effect on BTC than the rate of the pound.
Large UK and EU companies will make sure they are well represented in the negotiations for new trade agreements, so they will be much less affected. The UK was in favour of a trade agreement with China, but the EU has blocked that until now. I don't know how good this would be for the UK, it would definitely be interesting, but maybe not affect BTC much.
The remaining countries in the EU are facing a hard choice: a few countries will need to make up the membership fee of the UK, or the EU would need to cut spending. The first will mean speeding up the money press, the second has never happened before, a may cause trouble for some of the EU countries that are depending on EU cash to survive.