Do alkyl halides with an alpha nitrogen atom undergo SN1 reaction?
The reactivity of haloalkyl ethers are well documented (e.g., they were originally used to protect $\ce{-OH}$ groups during total syntheses). For instance, chloromethyl methyl ether had been used to protect alcohol and phenol $\ce{-OH}$s as $\ce{-OCH2OCH3}$ or commonly known as $\ce{-OMOM}$. They should be sealed and stored in freezer to protect from moisture because they are susceptible to decompose by reacting with water, showing their reactivity. They are also well known carcinogens. An early review on $\alpha$-haloalkyl ethers can be found in Chemical Reviews (1955)([Ref. 1]).

A recent example in Ref. 2 shows in situ preparation of $\alpha$-halo ethers to protect $\ce{R-OH}$ groups, in order to prevent researchers from exposure to carcinogenic $\alpha$-halo ethers (see Figure below):



For example for carcinogenic activity of a $\alpha$-halo ether: Human health assessment information on bis(chloromethyl)ether can be found on the IRIS website (https://www.epa.gov/iris). IRIS stands for Integrated Risk Information System, a part of EPA’s mission to protect human health and the environment.

Based on the high reactivity of  $\alpha$-halo ethers, one can expect $\alpha$-halo amine would have been even higher reactivity than them. To my knowledge, they do not exist, mainly because, they'd undergo elimination quickly to give relevant  Schiff bases (see below):
$$\ce{-&gt; [R-CH(Cl)-NHR1] -&gt; R-CH=NR1 + HCl} \ \mathrm {or}$$
 $$\ce{-&gt; [R-CH(Cl)-NR1R2] -&gt; R-CH=N+R1R2Cl-}$$

[Ref. 1]: The alpha-Haloalkyl Ethers: Lawrence Summers, Chem. Rev., 1955, 55(2), 301-353 (DOI: 10.1021/cr50002a003).

[Ref. 2]: Simple, Rapid Procedure for the Synthesis of Chloromethyl Methyl Ether and Other Chloro Alkyl Ethers: M. A. Berliner, and K. Belecki, J. Org. Chem., 2005, 70(23), 9618-9621 (DOI: 10.1021/jo051344g).