Ask for advice in your bookshop or toyshop. There are loads of games out there for 2 year olds, at least there are here in Germany. Children don't start school here until 7, and any attempt to teach them letters or numbers before then will result in a scathing from the kindergarten teachers. At least, that was what it was like when my boys were little.
So they have to be occupied somehow.
The Ravensberger Games also have instructions in English so you can probably get them in the UK.
Don't expect a 2 or 3 year old to play the game as it should be played - they make up their own rules. You have to improvise and go along with that. The dice in these games for 2 year olds have an dot in a different colour on each side or a similar system, maybe a picture on each side to go with the theme of the game.
As long as they are learning to throw the dice or pick a single card and that this means they can move something on the board, that is enough for the time being.
A game may be "over" for them after 5 minutes, and of course, they will have won. Then they can help tidy the pieces away and be shown how to treat the games with respect.
There are very simple games if pelmanism available for pre school ages, or you can make your own. For example, if you can't draw you could take photos of six or 8 everyday objects and print out two of each, or just draw simple shapes in different colours. Here they might not understand the game at first, but it is fairly intuitive even for toddlers. Just play along and guide them gently.
There are also a couple of games which don't have a winner, but these are for the older children. Until they are about 7, losing can cause tears and tantrums (tell me about it!). So these games, where everyone plays against a mutual enemy, are ideal.
We have two, one where you all have to catch a thief, and another where you all have fire engines and have to out out a fire.
If you haven't got much time, a game of noughts and crosses will help them learn strategies (maybe about age 5 onwards), or for older ones what we used to call "boxes" where you draw lines of dots and take turns to join them up until you start making boxes by joining 4 sides of a square.