TTL records define the 'Time To Live' for a DNS zone file - this is in essence the amount of time it takes before a change to the DNS becomes live.
If you are changing the nameservers, setting the TTL won't make a bit of difference, because the zone file that you are setting the time to live for is the old one - it won't effect the propagation of a nameserver change. If you want a seamless transfer, I recommend the following:
1) Change the TTL on both sets of nameservers to 300 seconds
2) Wait for the new TTL to become live (time depends on the old TTL - usually 24 hours)
3) Setup the necessary DNS for the new VPS on the old nameservers and mirror on the new nameservers
4) Once the site and email is working as it should on the old nameservers, change the nameservers to the new ones.
This way, changing the name servers won't cause any disruption because both sets of nameservers will already have the correct info on them.
I quite frequently have to help customers move sites to us while maintaining zero downtime.
Additionally, I recommend ensuring that you change your mail settings to collect from the IP of the old server rather than the host name and then creating another account connecting to the IP of the VPS, after the migration, leave the old server operational for at least one week as some ISP's cache DNS locally and will ignore any changes you make and you may find that for a while some email still gets delivered to the old server