I've been in this boat recently. You will need to register with HMRC for 'self assessment' the year you are fully tax resident in the UK. Noting that the UK tax year started on 6th April you will probably be UK resident for the 23/24 tax year.
You will need to do both Irish revenue and HMRC returns going forward. I found it easier to do the Irish ones first then the UK ones. This is because you need figures from the Irish one to claim credit on Irish tax paid against any UK tax bill.
In practice I found I would pay 20% income tax on Irish rental income to Irish Revenue, then put the amount paid in as a credit on my UK tax return. Where the UK rules differ is on mortgage interest on BTL properties. In the Irish context you can 100% deduct mortgage interest expenses before calculating your tax liability, whereas in the UK it is calculated as a 20% tax credit (so not as helpful). In the end you probably pay slightly more in tax overall by having a rental property in Ireland while living in the UK so not an ideal situation unless your UK stint is time limited and not going to be open ended.
In terms of tax return, I found a number of online firms very helpful - TaxScouts and GoSimpleTax were pretty good at helping with Self Assessments for UK tax returns. Ultimately, no matter who you pay, you need to be over the detail of your situation in order to make the tax returns as effective as possible. They can only go on the data you give them.
Also note the new regime with Irish revenue for non-resident landlord withholding tax. This could be a real pain in the neck for you if you are a UK resident with an Irish property rented out.