I am thinking about purchasing a complete .50 Beowulf upper
for my recently purchased M&P 15 .223. Those uppers only fit mil-spec lowers. Is the M&P 15's lower mil-spec?

Mil-spec dimensions?... yes. Truly mil-spec?... No, as there is no such thing as a mil-spec AR made for the civilian market.

There are military specs for the ar-15. Most aftermarket parts are made to military specs due to the wide market for them and should fit ARs from most manufacturers. For example: there are only a few receiver manufacturers who stamp them with logos and features for the different AR companies. If the manufacturing process was not held to some sort of standard, their products would be in limited demand. The standard is military specifications. With that said, I have seen some products that needed a little "adjustment" to fit properly.
AR-10 products are another matter. There is no Mil-spec standards for this adaptation of the AR-15 platform so manufacturers have no exact standards to strive for. Therefore it is best to purchase the major components from the same manufacturer.

Alexander Arms beowulf.50 fits fine on my colt m4 lower. The upper pictures show it without the muzzle brake and the lower features it with a Witte machine tactical brake.

Colt would disagree with you. You can not be "Milspec" without being inspected and accepted by government inspectors.

American Rifleman | The MilSpec Definition

Also, while it is true that there are few forges that produce the upper and lower receiver blanks, each individual manufacturer is responsible for the final machining and engraving... Two manufactures can receive the same forged blanks from the same forge, like Cerro Forge, but can provide different levels of finished quality due to their own machining and QC standards.

I guess I should have said "standard" instead of "mil-spec." Anyhow, I got an email from Ruger in which was said that Ruger lowers are standard. On the other hand, Ruger uppers and bolt assemblies are not, I believe. The Ruger firing pin is not interchangeable with other brands' and I read somewhere (but I may be mistaken) that the carrier is non-standard, either. Maybe Ruger wanted to prevent basement tinkerers from trying to make the AR 556 into a full-auto gun. Maybe. I don't know enough about AR 15's even though I have three of them: a Colt Sporter post-ban, dehorned, the Ruger, and a S&W M&P 15. When I bought a Bushmaster upper with a 24" fluted barrel for my Colt, I had to buy an offset hinge pin because the Sporter's pin did not align with the new upper. I wish those who make AR 15's all remained within the same specs as everybody else's.
Thanks for the answers. I am waiting for a new credit card that will give me $250 if I charge on it $1000 the first month, and as soon as I get the card I'll order the .50 Beowulf with the fixed muzzle brake, a 10-round magazine, and a few boxes of ammo with the Hornady XTP bullets. I want to use this caliber on hogs and deer. I have been using a Ruger M77 in .338 WM but it's too long and cumbersome in a blind and all my shots at the deer I've harvested these past five years (9 whitetails) have been between 10 and 90 yards.

Most uppers and lowers made today are going to be made to the same dimensions. The days of Colt using different sized pins for their commercial rifles are behind us. You can take the BCG from your Ruger AR-556 and run it in your other AR-15s... As a unit, it will work. What you can't do is run the M16 firing pin in your Ruger, as the larger flange will get hung up in the notched hammer. Why Ruger used this setup when every other manufacturer including Colt who developed it for political reasons, has moved away from it is anyone's guess.

As far as the Ruger upper, it will connect to other lowers... The part that is "non-standard" is the gas block/front sight assembly and the Delta ring assembly. This has caused Ruger owners some headache with certain after market handguard and rail setups. Ruger tried to reinvent the wheel for market differentiation.

I did buy the A.A. .50 Beowulf upper and mounted it on a S&W M&P 15. It dropped right in and works perfectly. I was wondering whether I should have a heavier buffer and buffer spring, but A.A. said it's not necessary. I was going to mount a nice Leupold scope on it, but I've just lost about 10,000 dollars in the stock market because of the Coronavirus scare, and I'll have to wait until better times to spend money on "unnecessary" things. So I mounted on it an old Bushnell that I had on another rifle before I replaced with a Leupold a year ago, and I'll have to make do with that.
Folks, this epidemic (or pandemic) is serious business--take it seriously. I would have liked to go to the NRA general meeting but I doubt it will be held this year, what with the tornadoes that hit Nashville and the danger of contracting the disease in places crowded by people from all states--even those severely hit by this deadly disease.
Good luck to all. Avoid crowds, public transportation, medical buildings, theaters, stadiums, parties, political rallies. Shop for groceries at night, when there are few people in the stores, remember that a sneeze or cough will propel the virus up to 15 feet or more, and that it will remain airborne for up to 30 minutes. And wash your hands often and don't touch your face before washing your hands. Carry some sort of disinfectant in a bottle and rub it onto your hands often
Sorry about the O.T., but these are hard times.