If you are assigning everything a static IP address as you had before, I don’t see any reason why the e-mails wouldn’t continue working. Switching to a different ISP is not something most of the users will notice (unless of course you are switching from dial-up to high speed haha).
Rate the importance of the unit when reassigning IP addresses. That is to say, remember to configure the devices to the new network/IP address before full implementation of the ISP switch so that you will be able to access the devices after the switch and won’t have to reset the devices. This will decrease downtime. Just document everything!
You can go ahead and add a new MX record for the same domain name (I assume that you will be keeping the same domain name) with a higher weight for the new IP address before it is active. Be sure to change the lifetime on the records to a shorter time so it becomes “live” quicker when the new address is active.