In the event of my employer's business being transferred to new ownership, does my new employer have to recognize my trade union?

Yes. If the change of ownership is a relevant transfer of an undertaking then the Transfer of Undertakings ((Protection of Employment) Regulations 1981 (commonly called TUPE) will apply.

If you are transferred to the new business your terms and conditions of employment, including your continuity of service, are protected. The effect of the transfer is as if you were still employed by your previous employer. This includes any existing arrangements where the previous employer recognized your trade union and so those arrangements must continue.