Yes, anyone driving the nominated vehicle will be covered for breakdown as long as they are insured to drive it and have your permission to drive the vehicle. To be covered in any other vehicle they'll need separate breakdown cover for that vehicle.
If you want to cover you and your spouse/partner in any other privately owned vehicle while in the UK then you can add personal cover to your policy.
Please note: Personal Cover is only available as an optional extra when Breakdown Cover is taken separately. It's not available when adding Breakdown to your LV= car insurance policy.