If the stain is allowed to cure for some time without being touched, will it bleed?
Many stains will rub off unless covered in something, and oil-based stains in particular tend to sit on top of the wood rather than being fully absorbed into the surface as with waterbased and alcohol stains (which are chiefly dyes, by comparison oil-based stains are primarily pigmented).
So as I mention in the Comments you really have no choice but to use an overcoat of some kind after staining, as I'm sure the stain instructions specify. In addition to the protection afforded this usually makes the wood look much better too.
The reason I applied the top coat was because the stain was bleeding after drying up and would leave oily residue when touched
This indicates the stain hadn't finished drying1.
As with all things finish-related, when it comes to drying and curing it's necessary to wait as long as needed. When it's cool, and especially humid, oil-based finishes can take much longer than the instructions indicate to get dry enough to proceed2.
Alternately, should i just wipe on another coat of the oil based stain and leave it be?
No definitely don't do that. All conventional stains ("gel stain" is an exception, because it's varnish not stain) require direct access to the wood surface to work as originally intended.
What would be the best way forward to salvage the project?
There's a chance that the waterbased finish will never dry properly. Even if the tackiness reduces to tolerable levels or goes away completely there's a chance the finish will remain softer than it should have been if it had dried at something like its normal rate.
This is something you can confirm by contacting the manufacturer (often advisable if something goes amiss with a finishing product, although check that they don't have an FAQ that covers the problem already).
If I'm correct then you have no choice but to remove the poly, and by this I mean strip it off, not sand it off. Sanding is the worst way to remove finish anyway, but it's nearly impossible to effectively sand soft and sticky finishes because they almost instantly clog the paper.
Either way though, removal will undoubtedly damage your stain job I'm afraid. As it's difficult to impossible to do touch-up staining and get it to blend in it's usual to go back to bare wood (and here you will have to sand) so you can begin again, that is unless you're willing to try to hide the inconsistencies by going much darker, e.g. to a Dark Walnut, Tudor Oak or Espresso colour.
1 Oil-based stains can be slow to dry, it's one reason they're so user-friendly. They give lots of time to apply and futz about, but with the resultant downside that you have a relatively long wait after you're done until the next step.
2 Many instructions give dry times based on ideal drying conditions, where the humidity is relatively low and the temperature perhaps 70°F (21°C) which I personally think is laughably unrealistic, even for the US. In the UK humidity is basically never low, and it's rarely above 20°C, ergo stains and clear finishes over here are made for those conditions.... so it's not at all difficult to formulate finishing products that work well in a wider range of conditions.