Having just ordered a fatboy I'd also love to know how people here keep them nice
I put mine into an ultrasonic cleaner from time to time. If I can, I put a tiny amount of silicone grease on any exposed moving places I can get to, which are usually the linkages, hinges etc. in the top end of the razor.
I have in the past been tempted to use a regular dose of light oil of some sort (and I've tried many including gun oils, sewing machine oil, natural vegetable oils etc.) but I haven't had very good results. I find that most wash straight out of a razor in use, with soap to break them down, and make a bit of a mess in the washbasin. The best use I've found for oils is in major "de-clagging" operations, where I ultra-clean, put plenty of oil in, move it all about to free it; rinse and repeat. Lubrication is obviously good for metal to metal surfaces, but it doesn't do a lot if the space between is filled with soap or scale.
The two main enemies of these razors are limescale and soap build-up. Once these build up heavily in any TTO, especially adjustables, they are difficult to shift, except by some form of immersion cleaning (short of dismantling and reassembling).
No doubt there are some powerful cleaners/lubricants such as Young's 303 etc. which might clean out a clogged razor and lubricate it, but I haven't tried these. My ultrasonic cleaner will keep an initially-clean razor in good order, and, using the heater and additive cleaners designed for it, can, with a bit of perseverance, clean up some badly-clogged razors.