Your cleaner doesn't understand employment law.
Who advertised?
Did you advertise for a cleaner stating the hours of work and the pay? Did you draw up a contract of employment stating terms and get her to sign it? If you did, it's possible you're her employer. This means she'd have employment rights.
If she advertised for clients, decided her charges and when she's free to work, it's probable she's self employed. Especially if she has other clients and a company name. This means she has no employment rights other than those stated in her contract, which ideally she should have provided you with and got you to sign, although a contract can be verbal.
If she's self employed she's not entitled to holiday pay at all. If she takes time off for any reason, it's unpaid. She can decide when she works, for how long and whether to take a day off. If you decide to stop using her she's not entitled to notice period (unless you have a contract with her saying you would) or redundancy pay. She's not redundant if she's not an employee.
If she's your employee, she's entitled to holiday pay pro rata.
I don't know exactly how it's worked out, but if legally holiday pay is eg 20 days a year for full time (figure plucked from the air!) that doesn't mean she gets 20 Mondays off, because she's not full time, it doesn't mean she gets 20 shifts off either.
I've had a part time job working 4hrs a day, 5 days a week. Full time workers got I think 28 days holiday and I got an amount in hours that worked out to something like 15 shifts off. It's a complicated system.
Even if she's an employee, there's no legal entitlement to have particular days off, not even Christmas day and definitely not bank holidays. The employer decides what days the employee can take. Some companies the employee can request certain days, other companies the employer decides it all. I've known companies allocate each worker 2 weeks over the summer and they have no say in which weeks it is. Other companies close for 2 weeks over Christmas and New year and all employees need to save some of their holiday allowance for this.
Obviously if she only works Mondays, then all her holiday days will be Mondays.
There's no legal entitlement to be paid double for working bank holidays or any other holidays. That is a perk of the job offered by some employers to attract good employees. Just like they might choose to pay more than minimum wage, or offer a company car along with the job, or private health insurance etc
I've worked in shops and if you usually work Monday then you worked it same as normal whether it was bank holiday or not and there was no extra pay. That's the norm in a lot of industries.