You have to file an execution petition under XXI CPC before the same court that passed the eviction order.
The court will then issue warrant for delivery of possession (bailiff will come and evict the tenant with the police help, if required).
The tenant can of course file an appeal before the Rent control appellate authority (sub court). If he files a stay petition also along with the appeal the appellate court may grant an exparte stay if you are not present.
You can file a caveat petition by which the court cannot pass any interim order or stay without giving you notice and hearing you.
A caveat is valid for 90 days and renewable.
Whatever, after the expiry of one month period, you can very well file an EP, but immediately file a caveat petition before the appellate court to safeguard against exparte stay. If the tenant files an appeal you will get notice and can strongly oppose the stay by highlighting the long pendency and landlord's hareships.
If you feel that your advocate is not guiding you properly or scaring you with unusual effects, you can change the lawyer and proceed as suggested.
If the EP and also the appeal is properly followed up without any delay from your side, you can get both the cases disposed expeditiously