Worthwhile pointing out, here..... even if you actioned the Part P submission this morning, it won't get actioned by the scheme until Monday at the very earliest and the relevant BC office won't file it for days.
Sounds like this agent is a total knob-jockey and probably in cahoots with the other electrician in just trying to make you look bad.
True - though the NICEIC one does generate an online certificate you can immediately download now - they are available at midnight following the form being completed (Can't see a reason why that bit wouldn't work at weekends, since it's automated).
But in either case, the Part P notification has nothing to do with the legislation for landlords, so this is clearly a letting agent who either wants to look like he knows everything, or is being awkward because he missed out on a cut.
Having said that, there is the general requirement of a landlord to maintain a 'safe installation' at all time, regardless of the EICR requirements - so I guess you could stretch and say that they need to satisfy themselves that any work has been done to the legal standards (which would involve a Part P notification for notifiable work).
If the letting agent was working on behalf of the landlord, then it would be reasonable for them to want to know that contractors doing work were appropriately Part P registered, for example, and that notification would be happening within the allowed time. A polite enquiry or a check on the competent persons register should have resolved that (and could have been asked of the landlord since he arranged and paid)
That's very different from demanding documents that he likely wouldn't have a clue how to interpret anyway (and has no right to anyway since he wasn't involved)
Sounds very much like someone didn't get their cut from work and is unhappy.