Yes, you can! But currently only in one direction...
As detailed in this IET article from July, Eurotunnel (who own the tunnel) have fitted leaky feeder aerials, base stations and repeaters to both tunnels, and it only took 10 months to kit out. Eurotunnel are then leasing them out to mobile operators.
The North Tunnel, which is normally used for trains travelling from France to England, has been assigned as "France" and leased to French mobile phone operators. Service in this tunnel went live in July, and you can now use your phone to call / text / surf whilst travelling from France to England. It's using French mobile operators though, so UK phones will be charged roaming just as they do in France.
The South Tunnel, which is normally used for trains going from England to France, has been designated "England" as is to be leased to English mobile phone operators. However, it's not yet live, as the operators are taking longer to sign contracts... So, you can't make a call whilst going from England to France yet, but that is set to be fixed fairly soon.
The service from France -> UK, does work well though, I can personally confirm it from earlier! If you're on the phone as you enter the tunnel, your call will continue uninterrupted, and work fine for the whole tunnel. However, just before leaving the tunnel your call will drop, as that's where the French network ends. A few tens of seconds later your phone will sign-on with the UK network, and you'll be able to call again.
AFAICT, there is no mobile service offered in the third tunnel, the service tunnel, only the emergency radios work there.