Sorry to hear about this, doesn't seem like they know what they are doing!
For a change of IP address like this, what they should of done is first dropped the TTL for the DNS down to 1 hour or less. This should be done at least 48 hours before the move.
As I think has been mentioned, other nameservers round the net keep a cache of DNS info, to speed up resolving a site. Normally, the cache is updated ever 24 hours or so (but can vary depend on how the nameserver is set up). As a result, if you change your DNS it can be 24 hours or more before its picked up. During this time, the nameservers will use their cached info, so if your ip address has been changed they will direct traffic to the old ip.
This means that after the ip change, your site will appear to be down, but then different people will be able to see the site after different periods of time. After 6 hours for example, I could be able to see your site, but someone else couldn't, because it depends on the cache time of the ISP's nameservers.
By setting a low TTL value, it tells other nameservers to look at your site more frequently, such as every hour, and update their cache from it. This should be done 48 hours before the move, so that this instruction propagates to all external nameservers.
Next, you change the IP in your DNS to point to the new server. Although at first your site will appear down to everyone, the DNS name servers will re read the DNS info from your nameserver and update after an hour. This means that the downtime will be minimized, as the changes will be picked up quickly by external nameservers.
Once the site has settled, you can then put the TTL info back up again, as if you are not planning any futher changes there is no need for nameservers to read your DNS so often.
Thankfully, although you are losing traffic, it shouldn't affect your Google rankings. Google knows that servers go offline, so won't mark your site as dead, but will check back later and then should be able to see your site. Your site won't be down for long enough for google to think it no longer exists.
For a couple of days after the move, your server traffic will be down, but should steadily increase as more and more nameservers receive the new information. Unfortunately, post move, there is nothing that can be down to speed things up.
One useful tip when this happens, is that if you want to work on your site, but your ISP hasn't updated the DNS with the new ip address, you can manually do it yourself at your end.
Open C:\windows\system32\drivers\etc\hosts in a text editor, and at the bottom you'll see an example like:
127.0.0.1 localhost
Add in your new server ip, in the format IP address domain eg
213.123.111.12
www.mydomain.com
Save the file. (Note that you may need to set windows to show hidden files to be able to see the files).
Now open up the Command Prompt, (Start -> Accessories -> Command Prompt) and type:
ipconfig /flushdns
This will read in the info you added to the hosts file, and it will override any net domain servers. As a result, you`ll be able to get to your site even though your ISP's nameservers haven't updated.
Don't forget to remove the entry a couple of days later, as if the DNS goes down for some reason, you won't know about it
