I've hunted in all of the places that I can think of on the NS&I website (and tried a few site-specific Google searches) to try to see if there's any information about saving form data hidden away somewhere but, regrettably, there doesn't seem to be any way to do it.
However, going as far as I could get without actually opening an account, I found that there appears to be no queue to use the site at the moment anyway, so perhaps you might be able to get everything done within the 10-minute time limit if you try again now?
Alternatively, if your body clock (and/or work schedule) will allow it, why not try at, say, 3am? I'm naturally fairly nocturnal and I often find that the very early hours of the morning are the best times for accessing normally-busy websites.