I maintain it’s not an HMO?

It's a HMO. The fact there's only one AST is not relevant.

If sh'e in residence but not a tenant I suggest you get to a property lawyer pdq.

Graham is totally correct. Four tenants on an AST IS an HMO. Although probably not licensable.

You should get advice asap - if you accept rent from this person after the tenants have vacated you may inadvertently create a new tenancy with her.

We have a fixed fee telephone advice service with a solicitor here: https://landlordlaw.co.uk/openaccess_content/the-landlord-law-telephone-advice-service/

Did your new (silent) tenant sign any paperwork when she moved in ? A Deed of Assignment would have been sufficient to bind her to the terms of the existing AST?
If her rent in up to date and you have served notice (presumably a section 21 if rent is up to date) all you can do is proceed to apply for a Possession Order in the County Court. However the burden of evidence non fault possession is much stricter than for Section 8 (arrears) cases so you will need to provide strong documentary evidence.
Whichever route you take the process is very slow so better if you can to persist in trying to communicate and negotiate with her to leave amicably

It may be a licensable HMO if your local authority has an Additional Licensing scheme in that area. Check their website.

If you did not deal properly with one tenant replacing another, eg deed of assignment as suggested above, then you may now have a problem.

If a tenant holds over after the other tenants leave, then the tenancy continues and the full rent is still payable, (not her portion), however, she may have wriggle room if she's not a named tenant. As above, get proper advice.

As you have accepted rent and presumably not objected then this person, for all intent and purposes, is a tenant. If it's a shared tenancy I assume it's Jointly and Severally liable so when the rent is due, and if you have no vacant possession, you could demand the rent from all of them, or one individual, even the ones that have left, if you have their contact details. You could bill the one in there for the whole rent. I think in all of these cases they would quickly ensure vacant possession. You certainly need legal advice, as these days it seems so easy for tenants to get away with nonpayment for a considerable time. 

First, it is an HMO, so you will need to determine if it should have been licenced. That will impact your position severely if it IS licensable.

Second, assuming you aren’t required to have a license, the uncooperative replacement tenant has likely become a tenant through your acceptance of rent from her, so is a joint tent. You rightly state that on the giving of written notice, in the form set out in your tenancy agreement, by any one of the joint tenants, the tenancy ends FOR ALL tenants. This is just about the only instance where a tenancy is lawfully ended even when a tenant hasn’t left the property. Any tenant remaining is holding over and by tenancy law can be liable for double rent of the whole tenancy (not just a portion), (unless unusually your tenancy allocates a portion of the rent to each individual tenant). I need to look up the specifics, but I’m out & about just now. Strictly each joint tenant is jointly and severally liable as vacant possession hasn’t been given, but it is your choice who you demand the rent from. I would firstly write to all joint tenants setting out their legal position from the date that the notice period has ended. It is highly likely that they are unaware of the potential downside and the vast difference between holding over after a landlord has given them notice and after they themselves have given the landlord notice. Ensure you quote the specific regulations so that they can easily verify this with legal aide or Shelter should they choose to do so. I suspect this will change the attitude of the reluctant tenant.

Lastly if all else fails I hope you have the notice in writing and can show it was validly given because you will need to apply to the court for authority to instruct the bailiffs. I’ve never had to do this, so I can’t advise on specifics. Usually a clearer understanding of their legal position is enough.

Okay, I've now had time to look it up and it's Section 18 of the Distress for Rent Act 1737. "This provides that where the tenant refuses to deliver up possession at the time stipulated in the tenant's notice, the tenant must pay the landlord double the passing rent and such double rent must continue to be paid while the tenant remains in occupation." (quote from https://www.druces.com/doubling-your-rent-how-to-maximise-returns-when-tenants-hold-over-wrongfully/) I believe you must notify the tenants in writing of their obligations regarding the double rent first. You can then claim for this during your application for possession of your property.
Druces further states that "Landlords should ensure that recovery of double value or double rent...goes hand in hand with the process of obtaining vacant possession of the premises. If the tenant remains in possession of the premises following expiry of the tenancy and the landlord accepts rent from the tenant in respect of the tenant's occupation without taking steps to obtain vacant possession, a new periodic tenancy could be created, entitling the tenant to remain in occupation of the premises on the terms of the new periodic tenancy and, potentially, subject to the security of tenure provisions of the Landlord and Tenant Act 1954"
IF however, it turns out that you should have obtained an HMO licence, then I'm afraid I can't advise you as I've never had an HMO. The above still theoretically applies, however due to the possible penalties you may be liable to, you may choose to handle the situation differently as, should it ever reach court, or the tenant obtains legal advice, you may find yourself in a world of financial pain.

As rent has been accepted from the new tenant it could be argued that a prima facie tenancy has been created but if no deed of assignment was drafted and signed this new tenant would not become a party to the original multi party tenancy
There may be the existence of a new 2 party tenancy between the landlord and that particular tenant only
If the notice was served severally it would thus not be valid on that new tenant (as he/she would not be party to the original tenancy unless that Deed was drafted and agreed by the remaining parties )
In such a case a new single notice would need to be served and in the absence of a written tenancy agreement possession could not be perused on S21 grounds
The lack of a license does not proclude possession on other grounds (ie arrears but there are none in this case) or the desire to sell the property etc. but full possession via a hearing would need to be sought, which in the current circumstances will be very slow

I am sceptical about whether an existing tenancy can be amended (presumably by way of re-grant as in the case of Sturgiss & Anor v Boddy & Ors https://landlordlawblog.co.uk/2021/07/22/tenancy-deposits-for-rented-properties-with-disorganised-comings-and-goings/) by accepting rent from another occupier to add a new tenant - particularly as many tenancy agreements have clauses saying that if rent is accepted from someone other than the tenants this is deemed to be paid as their agent.

Re the reference to Section 18 of the Distress for Rent Act 1737, Note that this is for the situation where the Tenant gives notice to the Landlord (not the other way round) and the Notice is accepted by the Landlord then the tenant doesn't vacate.