First of all, I should give the disclaimer that I'm not a historian, but I've read several books about the Tudor court. One book you may want to check out is called Henry VIII: The King and his Court, by Alison Weir. Weir traces Henry's health by looking at his armor measurements. Since armor was uniquely made and fitted for him, it's a good indicator of his size at the time. Here's the timeline she gives:

- Armor made in 1512, when Henry is ~21: 23 inch waist (pg. 5)

- Armor made in 1520, when Henry is ~29: 25 inch waist (pg. 223)

- Armor made in 1536, when Henry is ~45: 37 inch waist (pg. 429)

- Armor made in 1540, when Henry is ~49: 54 inch waist (pg. 429)

Keep in mind that in 1536, Henry suffered a jousting accident that resulted in a leg wound that never really healed. After this accident, his mobility was severely impaired and likely led to a major weight gain. Weir also notes that people at the time referred to Henry as fat. One direct quote is "the King was so fat that three of the biggest men that could be found could get inside his doublet" (pg. 449). The same page mentions that royal doctors, in 1542, thought he "was not of the constitution to live long" (Henry died in 1547 at age 56).

As to whether a waistline of 54 inches would be considered obese today, the NIH considers a waistline above 40 inches in males to indicate "higher risk for heart disease and type 2 diabetes." Therefore, Henry's measurement of 54 inches in 1540 would mean he was unhealthily heavy by modern standards too (https://www.nhlbi.nih.gov/health/educational/lose_wt/risk.htm).

I didn't find anything about horses being unable to carry his weight, but maybe some of the experts here will be able to elaborate further.