A bit of relevant info re whether or not ESCAPADE launches August next year as planned:

Rob Lillis (the principal investigator for ESCAPADE) has said "It hasn’t launched yet and we are concerned about that. But, having seen the Blue Origin facility at Cape Canaveral, I was much less concerned after seeing all the work they’ve done. I’m confident they will likely be ready for the launch of ESCAPADE."
Source: https://spacenews.com/escapade-confident-in-planned-2024-new-glenn-launch/

And while the above quote honestly doesn't make me super confident they'll meet this year's launch window; it's worth noting that even if ESCAPADE doesn't launch till 2026, there's a decent chance they'll still beat SpaceX simply because SpaceX will be focused on Starlink, recovery & reuse, HLS, Dear Moon etc; and unless they get a contract for a Mars mission (which don't come up very often, NASA have only sent a total of 3 missions to Mars (orbit or surface) in the past 10 years) they simply won't have any reason to send a mission to Mars (the last we've heard about Elon's plans for Mars is an uncrewed landing in 2027 which even if doesn't get delayed would still be after ESCAPADE).

Which means the bigger question (for me at least) is simply whether the ESCAPADE launch is successful when it does eventually happen, because if ESCAPADE is lost it's unlikely Blue Origin will have another opportunity for a mission to Mars for a while after.

So based on the article I guess the question is whether a spacecraft launched on New Glenn (like EscaPADE) will reach Mars before anything launched by SpaceX?

Or does it have to be the New Glenn upper stage itself?

@dp9000 Yes any payload launched by New Glenn would suffice.