I am so glad you asked this question directly. There have been dozens of "How do I get the stains out of my Simonis wool cloth?" but I never jump in because you already have dozens of members answering who think their Simonis is some kind of tender, sensitive fabric that should never be disturbed by cleaning solutions.
I have been repairing, restoring, and reselling vintage Singers for about 25 years as a hobby, and as a result, am a pretty good, but nowhere near great, craft sewer.
Wool is one of the most durable and long-lasting organic fabrics on the planet. Wool socks, despite going through wash after machine wash, will outlast any cotton sock 10 times over.
Please read this simple wool fact sheet before continuing:
https://d1cqrq366w3ike.cloudfront.net/http/DOCUMENT/SheepUSA/CharacteristicsOfWool.pdf
Google Characteristics/Features/Chemical Attributes/Uses of Wool (or any such thing) and you will find hundreds of references to wool and the wool industry.
One fact you will see repeated over and over again is the following:
A wool fiber can be bent back on itself more than 20,000 times without breaking, compared to about 3,000 times for cotton and 2,000 times for silk.
The national and international wool industry touts wool's structural durability while the pool industry continues to hoodwink and bamboozle the buying public into believing it's some kind of soft, delicate, sensitive fabric for PR purposes (in other words, it's better to buy a new one than restore it--just replace it if it gets too dirty or stained).
Like any fabric, water and heat can cause it to shrink, etc. But due to its coiled protein structure and its external scales, it has a resilience not to be found in something like cotton.
Below is a picture of my Simonis, about 4 or 5 years old, with about 10-15 hours of use per week on it, that I have personally flooded and washed about 4 times.
You can see the wear if you look close and compare it to the Simonis sample sheet, but nothing that is as a result of the washing.
View attachment 455287
By the way, I am going to replace it this summer--that's a nicer section without jump divots, etc. in it.
My method: Probably very similar to Jasonlaus' method. I take a lukewarm bucket of water with the appropriate amount of Woolite or similar in it and completely soak the table. Slate I believe (but am not certain of this) does accept a certain amount of water, but that will be dried along with the cloth. Use as much water as is possible without letting the water get absorbed under the rails (I roll towels and stuff them under the rails to absorb the water I don't want going under the rails.)
And here's the shocking part--grab your chair, Az'ers--you may fall off--you can gently scrub the cloth until clean. Wool is a really freakin' strong fabric. Again, think of those wool socks that withstand a good thousand washings before deteriorating. And, by the way, wool can now be milled and treated to withstand machine washings--amazing!
Let the water soak awhile, depending on how bad the staining is, and then use clean dry towels to absorb as much excess as possible--just let 'em lie on the surface for awhile. Then use fresh warm or cold water to rinse until you've thoroughly rinsed the soap out. Absorb the excess and put a fan to work blowing over the surface of the table until dry--I keep the fan on for a good 18-24 hours.
There should be no water marks or washing residue because you've done the whole surface and it's been evenly cleaned.
You're done. The wool will shrink and seem tighter right after the drying process, but due to it's inherent stretching capacity it will pretty much return to where it was before the washing. So, if it was loose before the washing, it will return to the same, or approximately the same looseness as it was before the washing. Read the wool info on the stretching and shrinking characteristics of wool on some of those websites if you need to understand why that happens to wool.
By the way, I play sloppy pool. That, and I use my pool table as a surface for preparing sewing and work projects, so I am not gentle on my table at all and it needs cleaning more than most other player's tables do. You pay me $300-$500 for some custom draperies and your draperies will end up starting on my table, too.