Book now or in the Fall?

It's very difficult to know which way pricing will go, so if fully refundable then book now and watch the prices. If you book directly with a hotel, perhaps start a little rapport and of prices drop, then ask for the new price.
Be sure you flag your calendar for when the refundable date ends. Europe is unlikely US chain hotels and some are only refundable for a time...sometimes 48 hrs prior to check in, sometimes several weeks.

In the current situation, your guess is as good as mine. But historically Eurostar tickets have been very cheap when bought in advance and very expensive if bought the week before. So if you can book a refundable train ticket you should probably do that.

I don't know what type of hotel, and what price range, you're interested in, but the Premier Inn chain in the UK usually has rates that are considerably lower if you book well in advance. Premier Inns are well-run and well-maintained, but they are not charming. I call them "business hotels", but they are a good value--though one of their advantages (usually being air conditioned) will not be of importance to you in October.

You should book your hotels now or more than 4 months in advance. Others are right to book only refundable hotels, even if the cost is higher. Write a note to yourself in your date planner book regarding the deadline to cancel and checking to see if travel will be allowed and whether you have to cancel. Don't worry about last minute deals. Its totally not worth the risk. Avoid the the risk of waiting until too many hotels are filled up and your choices are too limited.

A lot of people booked “refundable” hotels, tours, etc and didn’t get refunds in 2020. The economic fallout from this pandemic is far from over. For the best deals booking early is usually best, but if you do be fully prepared to lose every dime and not cry about it.

Plan it but don't buy anything more than a month in advance

The last two weeks of October are half term holidays for many schools in the U.K., so places will be busier and therefore more expensive. Booking last minute may mean very little choice, particularly as we have been advised not to book foreign holidays and take staycations.

One very interesting article I read mentioned that the author anticipated that things will fill quickly once a country opens to international travel due to an incredible pent up demand.

I can relate to that incredible pent up demand........

I have used booking.com to find and book lodging in Germany, Austria and Switzerland. Unless you take their usually slightly lower rate, you can cancel up to a couple of days before the reservation. A few times I have either taken the lower rate, or booked directly with the hotel. In the future I think I will always go for the possibility of cancellation. In that case, my credit card is not charged in advance and when I check in I pay with the local currency. Works for me.

What Joy said. Last couple of years we always go for the fully cancellable, domestic as well as abroad.