Brief description of a low degree of freedom shimmying wheel model is presented where the time delay effect is in the focus of the corresponding stretched-string-like tyre model. The stability charts obtained by linear stability analysis present various bifurcation phenomena. These are checked by experiments on a test rig and also by numerical simulations that involve the partial sliding of the tyre contact region as a nonlinear effect. The sense of the Hopf bifurcations are compared to various shimmy models including the classical single-contact-point ones. Double Hopf bifurcations leading to quasi-periodic oscillations are also investigated. The applied numerical methods are optimized for convergence and also for possible application in realtime control strategies.