Hi
Thank you for your question.
Based on the information provided in your question, it may be worth taking a look at some guidance attached regarding the Non-Resident Landlord Scheme which registration would be required:
What the Non-resident Landlords Scheme is
&
Non-UK resident landlords.
As the NRL scheme is created for people residing outside of the UK, the letting agent would deduct tax from your UK rental income and pay this to HMRC.
Alongside the SA100 Tax Return, the SA105 UK Property & SA109 Residency supplementary pages should also be completed to claim your personal allowances, as well as on your wife’s if she qualifies for personal allowances as well as disclosing any property income you receive.
To answer your last question, as the property is owned by yourself and your wife, we would expect that income from property and any expenses incurred would be split between yourselves.