No major issue. AT&T U-verse is designed to displace existing service. It can reuse Coax just fine. The installer will test your Coax to determine if it is in good condition. Your current Coax TV splitter can be replaced with a diplexer that looks almost exactly like a CATV signal splitter.
The signal routing process is a little different in that the U-verse VDSL signal runs from the outside NID location to the where your Gateway is installed and then the IPTV is backfed using the same Coax to the diplexer to then distribute the IPTV signal to your other rooms.
Often after checking the existing Coax the installer will only need to replace the existing Coax connectors with new compression fitttings at the connection points and at the existing wall jacks.
I suggest finding the IP address of your ATT gateway. You can find that in the Settings>Broadband>Status. Look for the item called Default Gateway. Write down that IP address.
Then ping that over long periods. I suggest the standard version of Ping Plotter. It has a 30-day free trial. It will graph your connectivity vs time, and that can help describe what you are seeing. If there is a time pattern, you could spot that too.
There are other ping programs that you might like.
Before we start talking about a RG do you know what kind of install you are having done? If you are a long distance from the cross box you may be getting an iNID and a different inside unit.
In this case the TV would be run with coax to the diplexer. Phone will also be tied in to this outside box.
Inside you will have you inside unit which the computers hook up to (this unit is ran over a single pair of phone cables to the outside unit) and a power supply with a backup battery run to the outside unit.