Yes. L-2 spouse status carries employment authorization incident to status regardless of the spouse's nationality, so a British citizen married to an L-1A transferee may work for any U.S. employer. He may need a visa stamp from a consulate, unlike a Canadian citizen.
Nationality changes the paperwork, not the work rights
L-2 spouses are employment-authorized by operation of law, and that applies whether the spouse is Canadian, British or any other nationality. The practical differences are procedural. Hypothetical example: a Shannon logistics manager's husband holds a British passport and Canadian permanent residence; a Canadian-citizen spouse could seek L-2 admission at a port of entry with the approval notice, but he will usually need to apply for an L-2 visa at a U.S.
consulate first, which adds an appointment and its own timeline. Once admitted, his admission record annotated for a spouse is what employers rely on as evidence of work eligibility. Children under 21 may study but not work and lose L-2 status at 21.
If a child holds only British citizenship and Canadian PR, the same consular step applies. The whole family's status is tied to the principal's, so the employer's plan for the Texas role, and its seven-year ceiling, sets everyone's horizon.