This should really be in pre-1900, unless we're talking a particularly interesting TL here...
We need to be very careful about mentioning the Levellers as a discrete entity at this point- hell, we even need to be careful calling them Levellers, as it was only later that Lilburne embaraced the term and as of 1647, their best chance for power, it was still being used as an insult.
Your best bet of getting the Levellers in power is actually IMO getting a Royalist Victory in 1643/44, after which Charles makes himself obnoxious to all and sundry, radicalises his opposition and eventually goes down to a popular revolution.
Absent this, you need to have the Corkbush Field mutiny go badly wrong- or right. Have somebody panic and shoot Cromwell and Ireton as they ride over to harangue Lilburne's Regiment, and then have the mutiny balloon, in a sort of 'in for a penny, in for a pound' sort of way. It's quite possible.
Only problem is that you've got a mutiny then, not a revolution; the Army (asuming the Army mutinies and doens't put down the rebellious regiments) can march on London and probably take it, with the help of sympathetic supporters in the City, but then Parliament and King (not to mention the Scots) will probably team up against them. Can they win? Maybe, but it'll be a tough fight- and given the complete lack of cohesion in the 'Leveller' movement as mentioned above, the result could be absolutely anything really.