When did people first recognize the estimate sizes/scales and nature of celestial bodies?

The Great Pyramid of of the African Ancient Egyptians embodies an abundance of mathematical measurements related to the sizes and scales of astronomical objects, perhaps far too many to list in a single answer, though we will do so if called upon for further evidence. For example,

The Great Pyramid weighs 6 millions tons More than 2.5 million individual blocks used

Apparently built in 20 years cough (sounds like bullshit and here's why)

For this to be possible the ancients would of had to cut, transport and place 14 stone blocks per hour, 24hrs a day for 7,300 days. There are dozens that weigh at least 70 ton; that's 11% the mass of the Christ the Redeemer Statue in Brazil.

The largest granite stones in the pyramid, found in the "King's" chamber, weigh 25 to 80 tonnes and were transported from Aswan, more than 800 km (500 mi) away.

It's orientated to true north, with the precision of three sixtieths of one single degree.

The average space between each block is 0.5mm

It's height (481ft) x 43,200 = the Polar radius of our planet (in ft).

Measuring the base perimeter and x 43,200 will give you the circumference of our planet (in ft).

43,200 is not a random number. It is derived from the precession of the earths axis. The earth wobbles around it's axis like a spinning top at the rate of one degree every 72 years. 72 x 600 = 43,200.

They have given us the dimensions of our planet on a scale defined by our planet itself, pretty smart right? A scale of 1:43,200. This blows my mind.

See, e.g., The Great Pyramid & The Cardinal Dimensions Of Our Planet: Ancient Egypt Part IV; The Great Pyramid of Giza a mathematically encoded structure; Pi and the Great Pyramid; The Great Pyramid Of Egypt Exactly Marks The Center Of Earth's Landmass

I think it’s also interesting to note that if you multiply the height of the pyramid by a thousand million, you are left with 98,000,000 miles, which corresponds approximately to the distance between the earth and the sun.

At 10,450 BC—and at that date only—we find that the pattern of the pyramids on the ground provides a perfect reflection of the pattern of the stars in the sky. I mean it’s a perfect match—faultless—and it cannot be an accident because the entire arrangement correctly depicts two very unusual celestial events that occurred only at that time.

First, and purely by chance, the Milky Way, as visible from Giza in 10,450 BC, exactly duplicated the meridional course of the Nile Valley; secondly, to the west of the Milky Way, the three stars of Orion’s Belt were at the lowest altitude in their precessional cycle, with Al Nitak, the star represented by the Great Pyramid, crossing the meridian at 11° 08’.8 -Robert Bauval

The Dogon of Mali in Africa, who are direct descendants of the African Ancient Egyptians, have a long history of astronomy, including knowledge of Sirius B, see About the Dogon; Dogon Tribe Nommo; The Dogon, the Nommos and Sirius B; The Pale Fox by Marcel Griaule, Germaine Dierterlen.

Thus, we can trace scientific estimations of sizes and scales of celestial bodies by African Ancient Egyptians to at least 10,450 B.C.E., the Dogon, who are descendants of the African Ancient Egyptians from 3200 B.C.E.; the science, culture, and traditions of the African Ancient Egyptians continuing today within the descendants of the African Ancient Egyptians across the Earth.

As Pieter Geerkens points out, even the very same Greeks and Romans whose original religion defined planets as gods believed that planets were indeed inanimated objects similar to Earth - at least those enlightened enough to study their properties. I would say that, at the very last, the moment christianity took over everybody believed that planets were not gods - or were burnt at the stake.

As for the relative sizes, only Luna and Sol are big enough to determine its angular size without special instruments - and even then you can fail spectaculary like Aristarchus - so you had to wait until the invention of the telescope to get measurable angular sizes for the rest of the planets, and then wait to the transit of Venus of 1769 to get an actual measurement of the Earth-Sun distance to convert relative distances to absolute ones.

I think, combining answers and comments here, three "milestones" can be established.

1. Estimate of the size of the Moon and Sun, by Aristarchus, as mentioned by Pieter Geerkens, gave a rough ballpark on the size of the two bodies. Heraclidean system gave the estimate distance to other planets, and by extension, a general clue about their size. (thanks to Pieter Geerkens)

2. As described in The Great Ptolemaic Smackdown - 1609:

TOF pauses once more to note that the real Scientific Revolution in astronomy was to move astronomy from a branch of mathematics to a branch of physics. Galileo intuited from his study of the Moon that the heavenly bodies were physical places much like Earth, about which physical discoveries could be made.

This was unrelated to size, but a breakthrough in understanding what the celestial bodies are - not some mysterious heavenly spheres of theological nature nature, but places with mountains, geography, etc. (thanks to Michael Seifert)

3. First rough, in 1639, with about 36% error, then much more precise in 1761 and 1769 - transits of Venus. This is where at long last the distance between the Sun and Earth was measured. Up to that point, astronomy was all about proportions; you could learn how many multiples the size of the Moon the Sun was, and how many Moon diameters was Mars away from the Sun, but the entire domain lacked the singular unit anchoring it in "earthly" units of distance/size. Knowing proportions between Earth orbit and all other celestial bodies, their actual sizes could be finally calculated, not just ballparked basing on very poor quality observations of Earth shadow. (thanks to Pieter Geerkens again)

Aristarchus is the first to measure the relative size of Earth and the Moon in the third century B.C., at a ratio of 8:3. This is a bit lower than the modern calculation at 3.7:1 due to ignorance of the penumbra shadow's narrowing.

Likewise Aristarchus calculates that the Sun's diameter is between 18 and 20 times that of the Moon. We now know that an observational error (due to limits of unaided human vision) results in this value being too small by a factor of about 20, but still clearly extremely massive compared to Earth.

So in answer to your question - since the third century B.C. the Greeks at least were aware that the heavenly bodies in our Solar Systems are of a size comparable to our Earth.

Here are more notes on the History of Geodesy, including for non-European calculations.

Note also that once the sheer scale of inter-planetary distances is known, it becomes self evident that the planets must be of comparable size to the Earth and Moon. That accurate calculations were not always technologically possible is different from understanding the obvious scale of those sizes.

This was a very slow and gradual process, which still continues, so the question "when" has no meaning here.

Ancient Greeks already knew that "The moon is larger than Peloponnese". But the first scientific measurement of Moon's parallax that we know is recorded in Ptolemy (II cent AD). Parallax gives you the distance to the Moon in terms of the size of the Earth, and the size of Earth was known, about 400 years before, though very approximately.

The correct order of distances (Moon, Venus, Mercury, Sun, Mars, Juliter, Saturn, stars) was known from the times immemorial, except the order of Mercury and Venus).

(Aristarchus made the first attempt to make conclusion of the ratio of the distance to the Sun and the Moon, but his attempt was unsuccessful).

The relative distances of planets were roughly known since the late 16th century, when the correct kinematic model (Copernican) was accepted, and Kepler's laws were discovered. Speaking of the distance to the stars, Copernican model only implies an estimate from below. The parallax of the stars was measured for the first time only in 1838.

Speaking of more remote objects, than the closest stars, no direct measurement of parallax is possible, but there are other methods (I recommend the very clear exposition of this by Hawking, Brief History of Time). The distance to the most remote objects is estimated using the Hubble Law.

Once you know the distance to the object and compare it with the brightness, you can make conclusions about its size.

Literature to begin with: Wikipedia, "Parallax".