To preface - I have only ever done anything like this with guitars, but my opinion is that they probably require stricter "protocol" than a piece of furniture.
When I moved my old/nice guitars from dry (30% humidity) to where I live now (~65%), I was told by a friend who works in instrument restoration, to step up the humidity by keeping them in a closed room with a humidifier, and to only step up the humidity a bit every week. ~5%. One of them is sentimentally valuable, otherwise I'd probably have just turned up the humidity and sorted it out later.
At the very least, the gradual nature of doing it that way, combined with checking it weekly, will ensure that you catch any cracks or issues and have a chance to figure out a next step, rather than crossing your fingers and going from British basement to dry room.
Whether it's worth sequestering and buying a humidifier, up to the end user I suppose in this case