Bicycle Mechanics - Reynolds Alta Race Wheels- Can't get tires on

Anyone else have a problem with this? The wheels are made for 700c tires.
Ive got a couple of Conti Ultra Race 700c x 23 tires, and I can't
get the last bit of bead over the edge on both wheels.

Contis are well known for being hard to install.

You'll need help pinching the tire beads together opposite the side of the tire your trying to get the bead over. You also need to keep the beads held in the center of the rim. I made a tire holder using some 1/2" EMT conduit with alligator clamps attached to it in several places. The EMT conduit is bent in a radius such that the clamps are held in just the right place to clamp the tire beads together around the rim allowing the tire bead opposite the clamps to be pulled over the rim. One or two people can do the same thing. A 6-pack of beer for each person makes getting help easier.

Im not looking forward to changing tires on the road.

I had to put on conti attack/force on those rims for a customer and they were on the tight side. Use the palm of your hand to push the bead over the rim rather than trashing your thumbs. The bead should stretch a little bit once you have air in the tires so changing them on the road shouldnt be too much of a problem

I dropped this bike off at the LBS for various work to be done.
I asked my favorite mechanic about this problem, and he demonstrated
in about 10 seconds how to do it. He took the wheel off, put the end
that was partially off the rim on the floor. With a hand on each side of
the wheel, he startedat the top and stretched the tire bit by bit towards
the problem spot. When he got to that spot, he had added enough slack
to be able to pull the bead onto the wheel. I was amazed.