The most common and most likely answer to this crossword clue is the 8 letter word LEVELLER
I've seen this crossword clue in The Times
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More clues leading to the result LEVELLER
Buddy explains!

The straight part
We know that "Old radical populist" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case old radical populist = leveller.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does LEVELLER mean?
LEVELLER adjective- The same height at all places; parallel to a flat ground.
- At the same height as some reference; constructed as level with.
- Unvaried in frequency.
- Unvaried in volume.
- Calm.
- In the same position or rank.
- Straightforward; direct; clear.
- Well balanced; even; just; steady; impartial.
- Of even tone; without rising or falling inflection; monotonic.
- Perpendicular to a gravitational force.
- A person or thing that levels.
- (especially) Something that transcends people’s differences (such as social class, wealth, etc.); something that tends to eliminate advantages and disadvantages.
- A person holding a political opinion in favor of eliminating disparities between the haves and the have nots.
- An equaliser.