Hi OP,
Something I don't think anyone has mentioned in the "buy a rental idea" is tax.
You would pay tax on rental income after allowed deductions at your usual rate - so usually 20 or 40% assuming you are working and have used up your nil band.
So, for example, if you owned a property outright and received £600 rent and imagine £100 each month on average is spent on insurance, gas checks, repairs, maintenance and voids. The £500 remaining would be taxed and you would get to keep £400 or £300 depending on your tax rate (declared annually on your tax return).
If you buy a rental with a mortgage and are a higher rate tax payer you can't deduct mortgage interest in full (they get a 20% tax credit so their mortgage interest is effectively taxed at 20%). So if you are a higher rate tax payer you are not only taxed on your "profit" but on your mortgage interest expense too. It is possible as a higher rate tax payer to have a tax bill for a loss.
To decide if you will be a higher rate tax payer you have to add your rental receipts to your salary and other taxable income so even if you are not a higher rate tax payer now depending on your income and amount of rent received you could become one.
Then there is capital gains tax when you want to sell the property to buy your own. The gain in value between buying and selling is subject to capital gains tax less your allowance (because the property was rented out and not your home). The capital gains annual exempt allowance is reducing from 12 to 3k.
I agree you are better off staying where you are unless you can afford to buy a bigger property. If you do buy again make sure your solicitor protects your existing equity for you in the new property if it is held jointly. As you and your partner have your own children you might want to hold any new joint property as "tenants in common" rather than joint tenants so you can pass your share to your children on death rather than it passing automatically to each other.
The posters that talk about your own security as the homeowner in the relationship make a good point. If you stay put in 15/20 years time (depending on your mortgage length) you will own your own 4 bed home outright - and continue to own it if your relationship has broken down. If you sell up and rent with your partner in years to come you might own find your savings have ebbed away and that you can't afford to buy/can't get the length of mortgage term you need.