Cheapest is a fair question and it has two different answers depending on what you want the room to be. If the conservatory is a summer room that has got tatty, there are cheap fixes that work. If it has to become a room you sit in all year, the cheap options stop being cheap because they do not do that job at all.
The cheapest fixes that are not replacements
Resealing the glazing bar caps and renewing the end tapes on polycarbonate sheets cures most of the leaks and the stained cloudy panels for very little. Replacing the wall flashing with one properly dressed and pointed into the brickwork cures the leak owners most often blame on the whole roof. Cleaning and clearing the box gutter stops the mysterious heavy rain leak. Together those cost a fraction of any replacement and are worth doing first, because they also tell you whether the roof itself is the problem.
The cheapest actual replacement
A like for like polycarbonate sheet replacement is the lowest priced replacement, and on a roof whose sheets have gone brittle it restores the whole covering in a day without touching the frame's loading. It leaves the room exactly as seasonal as it was, which is the honest trade. A step up, and still well below a solid roof, is a modern solar control glass unit, which keeps the daylight, cuts the summer heat noticeably and again leaves the structure unchanged.
The cheap option that is not a replacement at all
Internal insulated ceiling systems are fitted under the existing roof rather than instead of it. They are genuinely much cheaper than a solid roof, they make the room warmer and quieter, and they leave the original roof above where you can no longer see it leaking. That is a real product with a real drawback, and it should be compared against a proper replacement rather than sold as one. If the existing roof is sound and dry it is a reasonable buy; if it is not, it is a lid on a problem.
Questions people ask about cheapest way to replace conservatory roof
Is it cheaper to repair a conservatory roof than replace it?
Nearly always, and the repairs that matter are small: the wall flashing, the sheet end tapes, the bar gaskets and the box gutter. Once two or more polycarbonate sheets have cracked with age, replacement usually wins.
Will a cheap replacement roof still need approval?
A like for like sheet or glass replacement normally does not, because the roof stays substantially translucent. A solid roof commonly does, which is part of why it costs more.
Do insulated ceiling panels stop a leaking roof?
No. They hide it. Fix the roof first, then decide whether you still want the ceiling.