Federated edge learning (FEEL) is a popular distributed learning framework for privacy-preserving at the edge, in which densely distributed edge devices periodically exchange model-updates with the server to complete the global model training. Due to limited bandwidth and uncertain wireless environment, FEEL may impose heavy burden to the current communication system. In addition, under the common FEEL framework, the server needs to wait for the slowest device to complete the update uploading before starting the aggregation process, leading to the straggler issue that causes prolonged communication time. In this paper, we propose to accelerate FEEL from two aspects: i.e., 1) performing data compression on the edge devices and 2) setting a deadline on the edge server to exclude the straggler devices. However, undesired gradient compression errors and transmission outage are introduced by the aforementioned operations respectively, affecting the convergence of FEEL as well. In view of these practical issues, we formulate a training time minimization problem, with the compression ratio and deadline to be optimized. To this end, an asymptotically unbiased aggregation scheme is first proposed to ensure zero optimality gap after convergence, and the impact of compression error and transmission outage on the overall training time are quantified through convergence analysis. Then, the formulated problem is solved in an alternating manner, based on which, the novel joint compression and deadline optimization (JCDO) algorithm is derived. Numerical experiments for different use cases in FEEL including image classification and autonomous driving show that the proposed method is nearly 30X faster than the vanilla FedAVG algorithm, and outperforms the state-of-the-art schemes.