Now look you lot this is getting technical and I am dredging this up from 40 years ago.
Yes its an accumulation
I think everyone who fancies their chances as a songwriter adds a line as an initiative test and I reckon variations must abound.
S&R has the form it does accumulate men.
Here is what I believe is the correct form tho saying that is risky.

One man went to mow
Went to mow a meadow,
One man and his dog Spot,
Went to mow a meadow.

Two men went to mow
Went to mow a meadow
Two men one man
and his dog Spot,
And a bottle of pop,
Went to mow a meadow.

Three men went to mow
Went to mow a meadow,
three men two men
One man and his dog Spot,
And a bottle of pop,
And a sausage roll,
Went to mow a meadow.

Four men went to mow
Went to mow a meadow,
Four men three men two men
One man and his dog Spot,
And a bottle of pop,
And a sausage roll,
And a big pork pie
Went to mow a meadow.

Five men went to mow
Went to mow a meadow,
Five men four men three men two men
One man and his dog Spot,
And a bottle of pop,
And a sausage roll,
And a big pork pie
And a knife and fork
Went to mow a meadow.

Six men went to mow
Went to mow a meadow,
Six men five men four men three men two men
One man and his dog Spot,
And a bottle of pop,
And a sausage roll,
And a big pork pie
And a knife and fork
In a plastic bag
Went to mow a meadow.

Seven men went to mow
Went to mow a meadow,
Seven men six men five men four men three men two men
One man and his dog Spot,
And a bottle of pop,
And a sausage roll,
And a big pork pie
And a knife and fork
In a plastic bag
And the girl next door
Went to mow a meadow.

Eight men went to mow
Went to mow a meadow,
Eight men seven men six men five men four men three men two men
One man and his dog Spot,
And a bottle of pop,
And a sausage roll,
And a big pork pie
And a knife and fork
In a plastic bag
And the girl next door
And Old Mother Riley and her cow,
Went to mow a meadow.

Well it looks OK and it sings OK
Bit of folklore
When I was very young every year during the summer the local working mens clubs organised a day out for their members and families. All the kids got sweets and money and tickets into fairs etc and hundreds of people then departed on hired buses to the seaside.At the time Grangetown in the NE of England was a fairly small place and the whole place was deserted on trip days At the seaside all the men disappeared off to get drunk and all the families ended up on the beach. During the ride there and back the kids would be in separate buses to adults and the song above is typical of the sort that was sung in the kids coaches. Anyone who had an extra line could stick it in and everyone kept singing till boredom killed it or the words ran out.
Never thought of this for years till now
How come Joe Offer that you and I only correspond on the more esoteric songs.

Fantum