Moo I have been in this position recently and I know how scary it can be. Have you told him to stop contact at least once? The legal definition of harrassment is two or more occasions of unwanted contact so if you have told him to stop and he's contacted you more than this then you have enough evidence already to go to the police.
I was told during my case though that the police dont give warnings anymore so you will have to decide whether to prosecute for harrassment. It may be different in your area though.
If you dont feel ready to go to the police, what I would do is block him on everything and (phone, email, social media) and if he bypasses this (e.g calling from withheld number, showing up at your house etc.) then rung the police. If he does show up at your house, do not answer the door and call 999.
He is very likely a narcissist and you have caused a narcissistic injury by ending the relationship. What he is doing at the moment is called hoovering (although what this actually is is harrassment) and hopefully he will give up fairly soon and move on to an easier victim if you remain silent and do not respond to any of his contact attempts. Brace yourself as he may move onto the smear campaign if you have caused sufficient narc injury (e.g. my ex tried to get me fired from my job by contacting my employer to make false allegations). I think it is unlikely he will try this given the relatively short relationship but you never know with these batshit dickheads. It's less likely (though not impossible) that he will become one of those horror story stalkers you read about and I'm sure he will give up soon and look for other narc supply.
It's really hit and miss with the police sometimes in cases like this but if they dont take you seriously, tell them how this is impacting you and that you are scared. The police are supposed to use how a victim feels about the risk posed an ex partner to assess how high risk your case is, along with other factors but dont let them fob you off (I've experienced this in the past). Ask to speak to a safeguarding officer who are specially trained in domestic violence.
I really hope this idiot gets the message and leaves you alone. Sadly he will just move on to another poor victim.