My daughter is seeing a tutor who has given her a method to do this - you learn the sequence, eg 4, 8, 12, 16 ... and as you say each number you put a finger down. You need to always use your fingers in the same order, so in the end you begin to associate the first finger with number 1 and so on. So eventually when you're asked 8x4, you know which is your eighth finger and you remember what number you normally say when you get there.
It seems to work! For 11 and 12 you put a hand on each hip, one at a time - that works really well.
The other resource her teacher has given her for learning tables is called "number trace" - you have several lines of numbers containing the sequence numbers in order but interspersed with other numbers (about 4 or 5 times as many). The learner uses a pen to underline all the numbers and circle the correct ones. At the top of the page the sequence is written up for them as an aide -memoire.
This is useful activity for learning times tables!
What's working is to practice the number trace and the finger method, then eventually moving on to asking random times table questions. The stage after that is to ask for a division sum with the same numbers.