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In the current paper, a generalized thermoelastic model with two-temperature characteristics, including a heat transfer equation with fractional derivatives and phase lags, is proposed. The Caputo–Fabrizio fractional differential operator is used to derive a new model and to solve the singular kernel problem of conventional fractional models. The suggested model is then exploited to investigate responses of an isotropic cylinder with variable properties and boundaries constantly exposed to thermal or mechanical loads. The elastic cylinder is also assumed to be permeated with a constant magnetic field and a continuous heat source. The governing partial differential equations are formulated in dimensionless forms and then solved by the Laplace transform technique together with its numerical inversions. The effects of the heat source intensity and fractional order parameter on the thermal and mechanical responses are addressed in detail. To verify the integrity of the obtained results, some comparative studies are conducted by considering different thermoelastic models.