'Do i start with solicitor and does anyone know what type of cost would be involved?'
Yes, start with solicitor. Will cost between 500-1000 ( plus VAT and outlays) depending on what you want to do ( new mortgage/ stay with old mortgage).
Ask for a general quote first but very often stuff shows up -( I am finding it next to impossible to complete anything at the moment with issues arising and banks being slow to deal with things) so better to go with a solicitor you've had recommended to you.
Its also very helpful if you identify what you want to do and when and if there are any issues ( planning, tax, boundaries, disputes etc.,etc - flag them early). So transfer into joint names and either continue with current mortgage or get new mortgage.
The better prepared and hands on you are the easier and cheaper it is for you/the solicitor.
mf