Triumphs use veneered dashes and I have worked on them a time or two.
There are glues that are supposed to work well with veneer. I tried traditionally liquid wood glue (Elmer's in overdrive) and was not happy with the result. It took too long to dry because when pressing to apply the veneer there was very little free surface area to breath and let the moisture out. I have heard some people swear by contact cement but I was too afraid to try that. One mistake with contact cement and you are done. Other people swear by Belhen type hide glue and an iron (not water based liquid). Were I undertaking this again I would give hide glue a try on some samples and scraps first.
As for the holes, veneer the whole dash first with no holes cut in the veneer. Have all the parts of the dash aligned and secured to a backing board so the grain lines up from panel to panel. Once the veneer is secured, use an Xacto knife to cut out the openings leaving a margin of about 1/8" to the edge of the hole in the dash panel. My method of final sizing and finishing the holes has been to use large diameter sanding drums and fine grit abrasive. Working at a drill press, lower the spindle/drum into the gauge opening and lock the spindle in place. Start the drill motor and work the panel around on the press table gently going around the circular cutout making multiple light passes. You will see and feel when the drum contacts the material behind the veneer.
When varnishing the wood, plan on many thin coats with light sanding with fine paper between each coat.
Above all, work gently and take your time. Practice with scraps a bit first to develop a feel for the process.