You can have two mortgages, yes, providing one is a normal residential mortgage (for the house you live in) and the other is a specific buy to let product (for the house you rent out). They can be with different providers etc and sometimes a bank will permit you to convert an existing residential product into a BTL if you have early repayment fees etc to consider. Residential mortgage affordability is worked out in the normal way based on your income/outgoings (normally around 4-5 x your annual salary), BTL affordability is based on a ratio of the rental yield (usually the rental income needs to be 125-150% of the mortage payments).
You will need to ensure you leave at least 25% equity in the rented out property as the loan to value rate on the majority of BTL products will be no higher than 75%. And of course you will need to make sure this leaves you enough capital to purchase your next house plus fees etc. E.g. if your current house is worth £150k, you have £100k equity, and you want to buy next place worth £250k, you could split the equity in half and get an 80%/£200k residential mortgage (if that's affordable on your salary), leaving £50k in current property and get a £100k/66% BTL mortgage (assuming that makes sense with the rental yield). But obviously if you have much less equity or are a lower earning making getting a sufficient residential mortgage difficult, it starts to be trickier. So you need to get going doing your sums, I'm sure people here will help you do rough sums if you can post: value of your current property and amount left on the mortgage, value of your new residential property, your approximate annual salary (and that of anyone else buying into the new house), expected monthly rental income for current property and any other savings/capital you want to invest into this project...
Also, bigger question really, why do you want to keep and rent out your current property? Are you experienced/knowledgeable about property management, have you done your sums with the true costs and worked out your profit margin, including all the things people forget like mortgage interest payments (the less capital you have the more these are), tax, agents fees, maintenance, void periods, tenants stopping paying rent and needing to be evicted, the list goes on... A lot of times these days for small-scale/one or 2 property landlords their BTLs end up costing them money in the short term so you need to be really sure it's the best use of your capital compared to other more passive/less stressful forms of investment.
If the reason for wanting to keep the property is to do with ring-fencing your money from an unmarried partner you're looking to purchase/move in with, there are other ways of doing this to make sure your money is secure, or if it's because you are relocating and want to keep a foothold in the local market likewise. Or if it's just purely as an investment, unless you have particular skills/interest in property management or think you've really hit on an opportunity for unusual growth it's likely there are better, easier, safer places to invest your money...