The reason I suggested a potential buy to let on your current home (or accessing a "Consent to Lease") is that this would effectively exclude any mortgage you have on this property from the affordability calculations on the new house that the lender would apply.
To give a rough rule of thumb calculation you would get 4 x income.
4 x £125,000 = £500,000
but if you take into account your current borrowuing this would mean you have a maximum borrowing of £170,000 for your new home and if you add to it your maximum £40,000 deposit this would mean that the maximum purchase price would be £210,000.
If you are able to exclude the home mortgage through buy to let / consent to lease and lets say this has a value of around £200,000 then then the affordability is a little more generous.
4 x £125,000 = £500,000 - £130,000 (second home mortgage) = £410,000 affordability.
Taking a buy to let would only be an option if you were no longer tied into your lender, it would be worth speaking to your lender and suggest that you may look to sell and do not want to tie yourself in for a long term and can they still offer you consent to lease. it may cost you £100 per year to access this (it may cost nothing) and of course you are not forced to let the property but it will give you access to a loop hole that would allow you to access a larger affordability for your new home. This is a short term solution only though as lenders will generally only offer CTL for 2-3 years maximum.
To go through your other questions.
Yes it is illegal to hold more than one residential mortgage as it is designed to be used against your main residence. You cannot have two main residences so to obtain a second mortgage designed to be a main residence is fraudulent and the lenders can impose some stiff penalties such as calling in the mortgage immeduiately or fraud markers on your credit file which will hagve a significant effect on the products and price if you try to obtain credit or any kind of insurance.
Your holiday home has a slightly different tye of mortgage against it. By design, when you bnormally buy a gholiday home it normally will state somewhere within the deeds that it is a holiday home and that it is not to be used as a residence for more than 11 months of the year and as such it cannot be a permanant residence, it could potentially be your main residence but it cannot be a permamant residence therefore it does not qualify under the lenders main residential mortgage rules althougyh often they will offer regular mortgage products to be taken on holiday homes. lenders often do quirte a lot of research on the properties before offering a mortgage as they need to make sure that any restrictiomns that may be placed on the property will not potentially affect the saleability of the property if you decided not to pay your mortgage.
There is no easy way around this. The above suggestion is just that, a suggestion, but it is legal and within the lenders criteria and will offer you a broader area to gain your mortgage within. In fact without taking CTL or looking into a buy to let you may find that lenders are not so willing to help you with another residential mortgage.