Have you tried playing the game offline and check if the items from Sim Delivery are there (in game)? If you do so, ignore the message saying "new content installed" and asking for a restart (I get it everytime I play offline even when there has not been any new Sim Delivery).
If the new items are nowhere to be found, try a restart (still offline) and check again. If they are still not there, go online and wait a bit until you get a pop up asking you if you want to restart or not, say no. Now go into your settings and check for new Sims Delivery, download if there is one, quit the game manually (not via pop up), wait a few seconds and start it again and see if the items are there in game.
For the "took too long to start" problem, I usually just wait a bit before trying to start the game again, in my experience this happens more often if I restarted due to the game asking me to, than due to me manually quitting (waiting a bit) and restarting.
Edit: since you are on xbox (like me) you can also try this, if you havent already, make this xbox your home xbox for your account, then log out of that account and create a new one. Log into the new account and open the game, download the sim delivery and restart, check if that worked. If it did, the new items should now be available for both accounts (that is how it works for me).