A lender offering a (mortgage) loan secured on a property will obviously require details of the property so the its location will be obvious to them. Some lenders may or may not offer loans (and perhaps different rates) for foreign property investments. If you mean remortgaging your existing property to cross finance the purchase of an investment property (domestic or foreign) then I guess its down to the lender as to whether or not they ask what the proceeds will be used for.
On most application forms you are asked for details of borrowings, other properties owned etc. Owning another property is actually something positive. If there is a lot of equity in your home it is positive. You should not have to tell your existing mortgage holder that you are applying for a loan to purchase an overseas property if it does not affect them in any way. However should you wish in the future to apply for a top up loan on your mortgage then you will have to disclose that you have purchased an overseas property. And as I said before this can be a positive thing especially if your overseas property has increased in value.
If you are borrowing against the family home to purchase an overseas property you will need the consent of your spouse. Most financial institutions will let you borrow for most things against the family home except a start up business and where you are entering in to a leasehold contract.
They would obviously prefer that you borrow for home improvements. This increases the value of your home and their security. A lot will depend on how much equity you have in your home and how much equity will be left after you have borrowed.
Buying an overseas property should not be any problem. As usual you will have to provide, current p60, payslips, certificate of income to show that you can afford to repay the loan. If you can get a letter from an overseas letting agent to show that you will also have an income from the property also helps.