No, you do not legally need a solicitor to write a will in the UK, but it is the most reliable way to make sure their wishes are properly upheld after death.
The best way to speed up the probate process is to be very prepared with all the necessary documents and to submit them to the correct authority without delay.
Yes, probate can be contested if you believe there is something not legal or ethical in the will, but this can delay probate from being granted even further.