It’s past midnight, so I don’t have time/energy to do a write-up with screenshots and stuff, but entries are now live in the new app, along with future entries. 
There may still be bugs to work out, but we’ll see how we get on.
I’ve processed future entries (in the new app!) from last Friday, last Tuesday, and today. And it should process them again tomorrow. You should get a notification for each run, telling you how many horses it entered, skipped (e.g. they weren’t picked this time for auto-entry), or errored (it will tell you why).
As usual, give a shout if you have questions. You can enter from the Race Schedules link on the left (for current entries), or from a horse’s page on the Entries tab (for current + future entries). Note that if you have settings for minimum criteria for race entries, these will apply on the race entry page (where you see Maiden, NW1 Allowance, etc and numbers of horses in those categories), but they won’t (yet) apply on the horse’s page. So if you have minimum energy B for race entry and you click to enter Bob in Saturday’s race, he won’t show up on the entry page unless he has A or B energy. I’ll be updating this to show a warning, or maybe not go to the race entry page and let you enter from Bob’s page. Not yet sure. 
@Andy you can schedule entries up to 6 months out, and you can have up to 5 races scheduled.