What's Concorde's engine fan blade tip speed at full thrust?
Apart from the fact that the Rolls-Royce/Snecma Olympus 593 engines were pure turbojets so they did not have any fan, it's pretty easy to calculate (better approximate) the tip speed of the first Low-Pressure Compressor's wheel.


The Olympus 593 had a diameter of about 1.2 meters [Wikipedia]. Let's say the first compressor wheel had a diameter of about 1 meter.
As we're talking about the LP Compressor, we're interested in N1 speed. At 100% N1 speed, we are at 6500 rpms [FAA Type Certificate No. E20EU] (for N2, at 100%, 8530 rpms).
Now it's just math to calculate, or, just use an online converter.


The result, as DoRoCro already wrote in his answer, is about 340 m/s.