I have done similar things in the past with at least fair, if not good to very good, success. Here is the key, no matter the variables...you have to get that tiny seed on the soil. If you broadcast into somewhat to very thick, tall weeds you are not likely to get a lot of seed down through the mess. If however, you broadcast into a tall, but thin stand of weeds, thin as in you can see the soil below, then you could be just fine. But most times, tall weeds are going to block your seed and severely limit your success.
Without being able to see your exact situation...I would probably do the following.
1. Mow it fairly short.
2. Broadcast your seed over the newly mowed field. Yes, there is going to be plenty of "duff" present that will catch a lot of the seed. Go a little bit heavier on the seed because you are going to lose some by not hitting perfectly bare ground. But...don't overdo it here, it is easy to get brassicas on too thick. I would add maybe an extra pound or two of seed per acre.
3. Spread your treated Urea. (Treated so as not to evaporate while laying on the surface of the soil. Don't use untreated urea if you cannot incorporate it into the soil.) Brassicas really need N, so don't skip this step.
4. KEY STEP HERE...drag the field...even two times if it is really thick duff on top. This is will "shake" the suspended seeds, and fertilizer granules, from the mowed grass residue that is likely to be blanketing the soil and cause them to drop through and get to the soil surface. You could alternatively mow the field again here in lieu of dragging. You just need a method to shake that tiny seed down through the duff and onto the soil surface. If you are going to mow twice, then your first mowing trip in step #1 above could be to mow it "medium" and then mow it short on the second pass.
5. Spray the remaining grass with gly to kill that off.
Ideally, you can get a nice rain or two on it just following this process. If you get that seed on the soil surface though, you should be golden. When properly executed...I would argue that the process above is just as good, perhaps better than, the normal process where the fertilizer is tilled in and then the seed is spread on bare dirt, etc. If you get those seeds down to the ground, then the remaining duff will act as mulch and hold moisture for you and keep the sun from baking the bare soil, etc.
I have done the above process, with slight variations, probably 5-7 times through the years...sometimes, like you, due to time limitations, but also, like this year again, because of trying to accommodate very dry soil conditions. Once again, my area is very dry now at this time of the year and conventional tilling can be a pretty rough go and tough on the tractor too AND nothing dries out the soil more than eliminating all of the residue and tilling it bare. If there aren't many/any good August/September rains then you can certainly get a brassica failure even when "doing it right" and tilling the soil and making a perfect seed bed, etc. The "throw and mow" method is more of a hedge against continued dry weather anyway.
