Outside of the US, you won't find fully rebuilt pianos with a few exceptions.
In Europe you can get ivory with a certificate to ship to anywhere in the EU without problem, but not outside the EU
...and there are quite a few exceptions now:
Piano Restorations Ltd in the UK do a full rebuild service, and are thoroughly able to do all makes. They're one of the official restoration centres for Blüthners. They replace soundboards, bridges, pinblocks, actions, keyboards, depending on what exactly needs done. All pianos they offer for sale are generally fully rebuilt with everything replaced including the soundboard. Steinways are rebuilt with Hamburg actions and keyboards. For comparison, a fully rebuilt Hamburg Steinway B will probably cost you something in the region of £40,000 to £45,000, depending on a few factors. Older "cores" are cheaper to buy, newer cores are more expensive, but honestly in a full rebuild you don't need to worry too much about whether it was a 1910 or 1970, you just need to worry if you like the piano. Blüthners and Bechsteins tend to be about £25,000 to £35,000.
There are a couple of other workshops in the UK offering a "full rebuild" service, but on closer inspection they're offering something more like a conservation service, and try to retain the original soundboard at all costs. Personally I don't have an opinion regarding soundboard replacement except to say my pianos have new boards and sound wonderful. I know someone on the forum is having a Steinway D rebuilt by a workshop who isn't Piano Restorations, and they ordered a Ciresa soundboard built to pattern for the piano, which the technician is installing. This is quite an unusual way to do it but not unheard of.
Steinway and Sons Hamburg offer fully rebuilt pianos and a comprehensive rebuilding service including soundboard and pin block replacement. Recently they've offered a few instruments for sale with the original board conserved, so technically it's more of a reconditioning than a rebuilding, but they are equipped to replace soundboards in the factory. A rebuilt Steinway purchased from Steinway will cost as much as 90 to 95 percent of the price of a new one. If the piano has a wood veneer like original Rio Rosewood, the piano is usually more expensive than a new black model, or at least that was the case a few years ago. They may have stopped this practice.
Blüthner UK offer rebuilt pianos and offer to rebuild pianos through two sources: their partners at Piano Restorations UK or through their Leipzig factory. The results are slightly different from each factory, Piano Restorations strives to restore the original sound of an early-mid 20th century Blüthner (and Colin has worked their long enough to have seen the new mid century instruments!), and the Leipzig factory tend to return the piano with a more modern voicing on it, but you can ask them to give it the traditional voice if you prefer.
The workshops in Poland such as SAP Renovation are growing in reputation and quality. They're also increasing in price. They will do full soundboard and pin block replacement. Sometimes their final preparation work isn't that good - or at least it wasn't a few years ago - but it's now at the stage where a good technician could regulate and voice the work they do and it would have a good result. SAP Renovation is probably the best Polish company.
C. Bechstein in Berlin now offer full rebuilding for Bechstein and other pianos at their Czech factory. This information is on their website. They will do a full soundboard, plank, action replacement.
Bösendorfer offer a full comprehensive rebuilding service. It's not on their website but you can arrange it through Yamaha London or presumably by calling Vienna directly. It might be that if you're in the USA, Bösendorfer would recommend going to Rich Galassini for a rebuilt Bösendorfer - and actually a trip to his workshop would be a good idea anyway if it's a rebuild you want to go for.
Steve Jackson is right that in many places in Europe including the UK, conservative restoration was always more common. Pinblocks (wrest planks in the UK!) were often retained, soundboards shimmed and filled and not often that well, and original actions fitted with new hammers. Of course there is a case to be made for retaining the original actions on some pianos, particularly those of a more historical interest (this post isn't about that anyway), but the industry is changing in Europe. More people are competing at a higher level than before, which is only a good thing for the industry. I've seen even the best rebuilders say they have to raise their game. Prices aren't cheap, but the quality is really very good.
Keyboards aren't always replaced when a new action is fitted, but many times they are, and even if the original keyboard is retained, having new non-ivory coverings put on is always an option for exporting instruments. Ivory pre-1970 isn't so difficult to export but you might want to have it replaced to save customs ripping it off in the port.
By the way if you can get a New York Steinway restored in the right way and set up by the right technician, it will be as good as a Hamburg, so don't rule out the NY Steinway particularly if you live in America anyway.
Personally if I'm shopping in that price bracket I'd also look at some new pianos. I wouldn't rule out a Yamaha SX series piano, for example, since they have a voice which is actually kind of similar to a fine rebuilt piano, but they have all the advantages of a new instrument. They're not cheap, but they're not the most expensive pianos on the market either so they're worth considering.