I went and purchased some bitcoins on my mycelium wallet. Once I went back to my car and opened the app to do a backup for it the app just stopped working and it wouldn't open back up. I couldn't do the back up to it, so I sent an email and asked if I uninstall and reinstall will I loose the coins that I just purchased? it took them an hour to respond to my email and by that time I had already uninstalled and reinstalled my wallet. The coins are gone but I looked up my transaction on Blockchain and they are there. How can I recover my coins when Mycelium had the glitch (that was fixed with an update) and there email was an hour late to respond to my question, how can I recover those coins?
If you do not have a backup of the wallet, either in the form of seed words or a exported file from Mycelium, then your coins are permanently lost.
Ideally, one should be backing up and restoring their wallet to make sure they've got all the necessary information before sending a significant amount of coins to it.
As others have pointed out, your money is irretrievably gone forever. There are some observations that might help you or might help future readers of this question.
I went and purchased some bitcoins on my mycelium wallet [later I] opened the app to do a backup.
Some people (maybe not you) have the idea that you need to make a backup every time your wallet receives money. This is not true.
It is superficially plausible, after all if the contents of the wallet have changed, surely the old backup won't know about the new contents?
The reason why this is not so is because Bitcoin wallets don't contain Bitcoins, the only important thing they contain is the secret number called a private-key. The globally replicated blockchain is what keeps track of who controls various amounts of money. You don't need to backup the blockchain yourself because every full-node in the worldwide Bitcoin network has a copy of it you can retrieve.
Immediately after installing a new wallet with no money in it, you can backup† the recovery/seed-phrase or the master private-key(s). After that, it does not matter how many times you receive money into the wallet, you can always re-create the wallet from the saved recovery/seed-phrase or private-key and it will recover control over all the unspent money received by that wallet since you made the backup.
There are some potential issues that are important to remember. If you import private-keys into a wallet, the saved recovery/seed-phrase or original private-key can't be used to recover control over money controlled by those imported keys.
I believe most wallets also allow you to make a full-backup the wallet which can be used to recover imported keys as well. So it may be prudent to make and test a full wallet-backup before and after making any significant change like updating a wallet to a new version, importing private keys, etc.
it took them an hour to respond to my email ...
Getting a response in less than a week from the developers of free software seems pretty good to me. If I was actually paying a monthly support fee I'd expect better than a week but a lot of commercial software support contracts don't guarantee a 1-hour response (and certainly not a 1-hour fix).
... and by that time I had already uninstalled and reinstalled my wallet.
That's a shame. If you feel a strong need to do something, often the best course of action is to shut-down and turn-off the computer/device and don't do anything with it until you have thoroughly researched and tested a solution using a different computer/device.
† write down on paper, imprint onto a metal sheet, etc. Put that somewhere very safe.
Yea it's a hard lesson to learn. The coins are gone forever but look at the valuable lesson you learned. Always copy those seed words down on paper and put them in a safe place. They are your private keys. I had a similar situation with electrum on tails. The wallet would not sync and I had $200 worth of bitcoin in it. I was able to regenerate the private keys to a different wallet with my mnemonic word seed and got my money back. So sorry this happened to you. It's not easy to be your own bank.