Hi can you tell me if the charger for my Apple Watch 5 will charge my new Apple Watch 7 with no problems?

Yes it should do but may not.

If it does not please do not think you have to buy a USB C charger to go with the lead in the box.

I have got a watch that has not charged with old series chargers since update 8.3

and I have the perfect solution rather than buy this charger.

I will see how this pans out with any other input and also you might bot have a problem at all

The 5w chargers charge from 4% to over 80% in just 36 minutes on my AW7